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NGN-LOCALLY CONVEX LINEAR TOPOLOGICAL SPACES
by
HORACE- NEAL PARKER, B.A.
A THESIS
IN
MATHEMATICS
Submitted to the Graduate Faculty
of Texas Technological College
in Partial Pulfillmemt of
the Requirements for
the Degree of
MASTER OF SCIENCE
Accepted
August, 1966
T3
19 U
No. 113
•-'' •> JA^.W*
Cop. 2
ACKNOWLEDGMENTS
I wish to express appreciation to my director.
Dr. Eugene Steiner, for his help and guidance in the preparation of this thesis and to my m6ther for the typing of
the final copy.
ii
TABLE OF CONTENTS
"^'^^
Page
ACKNOWLEDGMENTS
LIST OF TABLES
ii
. . . . "
"•
CHAPTER
•
I.
II.
III.
v
•
•
INTRODUCTION
X
GENERAL THEORY OP LINEAR TOPOLOGICAL SPACES
2
Introduction
2
Local Base
3
Continuous Linear Punctionals . . . . .
6
Norms and Metrics
7
Completeness
8
Category
9
Sups, infs, and lattices
9
Dimension
12
Miscellaneous
15
SPACES OF THE FORM l(Pj,)
Definition
l6
•
l6
Lattice of Linear Topologies
IV.
20
SPACES OP THE F0R14 L(p,ya)
26
Introduction
26
Lemmas on Measurable Functions
....
26
Definition of L(p,^
30
Relationships among the L(p,/*)»s. . . .
35
A Lattice of Linear Topologies
43
iii
....
-••••.
;•::;:•;.,.
-
i
v
Table of Contents (continued)
Chapter
V.
Page
PROPERTIES OP LINEAR TOPOLOGIES
47
Definitions . . . . . . .
General Results .
VI.
4?
*...
47
k-convexity
53
Preservation of Type
58
l(Pi), L(p,/<), AND S RELATIVE TO THE
PROPERTIES OF CHAPTER IV
65
Introduction
65
' (l(Pj,), u(Pi)) and (K-", w(Pi)
66
(L(p,/t), u(p)) and (L1, w(p))
70
The Space S
82
Two General Theorems, Table of Properties 84
LIST OP REFERENCES
88
LIST OP TABLES
TABLE
Page
1.
Summary of Theorems . ;
71
2.
Transfer of Klee's Properties
87
CHAPTER I
INTRODUCTION
Relatively little is known about linear topological
spaces which fail to be locally convex, but spaces of this
type do arise naturally in analysis.
In I96I Victor Klee
presented a paper at a symposium in Prague in which he defined several properties which a non-locally convex linear
topological space (l.t.s.) may possess.
He studied these
properties abstractly in some detail and cursorily mentioned several examples.
The purpose of this thesis is
to verify that Klee's examples do in fact possess the properties which he asserts and to find further examples.
One of the new examples which the author found is
a class of spaces studied by Simmons [7]. Besides the
properties defined by Klee, several new properties of these
spaces not mentioned by Simmons were discovered, and they
are presented in Chapter III.
The author found another class of spaces, not previously studied to his knowledge, which are closely analogous to the spaces studied by Simmons.
scribed in detail in Chapter IV.
They are de-
These and other examples
are discussed in Chapter VI relative to the properties defined by Klee.
Two special notational devices are used throughout
this thesis.
"If and only if" is generally written "iff"
and "///" marks the end of every proof.
CHAPTER II
GENERAL THEORY OP LINEAR TOPOLOGICAL SPACES
Introduction
This chapter will be devoted to a very brief description of some of the main aspects of the theory of __ -.
l.t.s.
We list a number of theorems of which direct ap-
plication will be made in the succeeding chapters.
For
a much more extensive treatment of l.t.s. the reader is
referred to Kelley and Namioka [2]; the numbers after some
of the theorems in this chapter refer to that book.
For
results about topological spaces in general see Kelley [1].
Definition.
A linear topological space (E,7") is
a linear space (vector space) E over either the real field
or the complex field provided with a topology ^relative
to which addition and multiplication are continuous in the
following sense:
if U is a neighborhood of x -f y, then
there are neighborhoods U and U of x and y such that
•^
y
X -f y e U^ -f Uy s U, and if U is a neighborhood of ax there
exist a neighborhood U^ of x in E and a neighborhood U^ of
a in the scalar field with the usual topology such that
ax € Ug • U^ s U.
• •3
•
*
.
•
Local Base
Definition:
A base for the neighborhood system of
0 is called a local base.
Definitions.
' ' "•''
A set M in a linear space E is radial
at 0 iff for every x « E, there exists a positive real constant b such that 0«a:«b implies ax c M.
A set M is
circled iff ax ۥ M whenever x ^ M and a is a scalar such
that la/ ^ 1.
A set M is convex iff ax -f by e M whenever
X € M, y € M, a ^ 0, b ? 0, and a + b = 1.
Theorem 2.1.
local base.
(1)
Let E be a l.t.s. and let 2^ be a
Then
for every U and V in ?^, there is a W in 2^ such
that W s UnV;
(2)
for every U in t£ there is a V in ZL such that
V -f V £ Uj
(3)
for every U in 2^ there is a V in U such that
aV £ U for each scalar a such that |a| ^ 1 ;
(4)
for each x in E and each U in E^ , there is a
scalar a such that x c aU;
(5)
for each U in E^ there is a circled set W and a
member V of W such that V £ W s U;
(6) each U c ^^ is radial at 0; and
(7)
if E is a Hausdorff space, then
n{U:U €?^I
= to}.
Conversely, let E be a linear space and U a non-void
family of subsets which satisfy (1) through (4), and let
T
be the family of all sets W such that, for each x in
W, there is a U in t^ with x -f U s W.
Bien J*is a linear
topology for E, and tt is a local base for this topology.
If further HlUrU e U\
= \o\,
then T i s a Hausdorff topol-
ogy.
TChe linear topology T of the last theorem is unique because it can be proved that in general given a linear
topology T and a local base ZC, a set 0 is open in J" iff
for each point x in 0, there exists ^5 e tt
X -f- U £ 0.
such that
That is, a local base completely determines
the linear topology, although a given topology may have
many local bases.
The idea of localization plays a very crucial role
in the study of l.t.s. We are generally much more interested in the neighborhoods of 0 than in the open sets
themselves. Many important properties are defined in
terms of the neighborhoods of 0, and certain topological
properties, like the continuity of a certain function, are
valid for the whole space if they can be verified at 0 or
at some other one point. One reason why the consequences
of localization are so sweeping is that the neighborhood
system at any point x is an exact copy of the neighborhood
9
system at 0; that is, it consists of all translations of
the form x -f U, U being a neighborhood of 0. Any translation
and any scalar multiple of an open set is open.
Definition. A l.t.s. is convex (locally convex)
iff there is a local base consisting only of convex sets.
Theorem 2.2. Let E be a l.t.s. Then
(1) the closure of a subspace is a subspace;
(2) the closure and the interior of a convex set are
convex;
(3) the closure of a circled set is circled;
(4) the family of all closed circled neighborhoods
is a local base; and
(5) if P is a subspace, E/P with the quotient topology is also a l.t.s.
If P is closed, E/P is Hausdorff.
Theorem 2.3. Let J^L ^^^ ^2 ^® "^^° linear topologies
on a linear space E.
Then J'^ ^ ^
iff every 7^—neighbor-
hood of 0 contains a ^--neighborhood of 0.
Note. We will say 7^ is finer (stronger) than 7^
iff J^ s J^.
but T^^
To indicate that -^ ^ ^
(i.e., that "^ ^
^
J'2) ^e will use "strictly finer" ("strictly
stronger").
Similarly for coarser and weaker.
Bieorem 2.4 (5.3).
A linear function f on a l.t.s.
E to another l.t.s. is continuous at each point of E iff
f is continuous at. 0.
Pieorem 2.5 (6.2).
A linear functional is continuous
iff the image under*f"of some neighborhood of 0 is bounded.
Continuous Linear Punctionals
An interesting and highly useful fact in the theory
of l.t.s. is that the nature of the continuous linear functionals on a l.t.s. reveals many things about the structure
of the space.
Definition.
A linear functional f on a linear
space E separates sets A and B iff f is not identically
zero and sup{r(x) : x € A } ^ inf {r(x) : x c BJ where r is
the real part of f.
f strongly separates A and B iff the
above inequality is strict.
Theorem 2.6 (l4.2).
Suppose that A and B are non-
void convex subsets of a l.t.s.
of A is non-void.
E and that the interior
Then there is a continuous linear func-
tional f on E separating A and B iff B is disjoint from
the interior of A.
Theorem 2.7 (l4.4).
Let A and B be non-void, dis-
joint, convex subsets of a convex l.t.s. and suppose A is
compact and B is closed.
Then there is a continuous
linear functional strongly separating A and B.
Theorem 2.8 (5.5).
If F is a dense subspace of
l.t.s. E and f is a continuous linear functional on F,
there is a continuous linear extension of f to E.
Norms and Metrics
A seminorm on a linear space E is a real valued
function II-II which has the following properties:
(1)
11011= 0 and U\\ ^ 0 for all x e E,
(2)
llx + yll € Jx// + Byll for all x, y 6 E, and
(3)
llaxll = laMlxlf for each x e E and each scalar a.
Any seminorm induces a semimetric d such that
d(x, y) = Ifx - yll. This semimetric topology is always
linear.
It is Hausdorff iff llx - yll = 0 implies x = y
(or equivalently //xj = 0 implies x = 0) in which case the
terms norm and metric are used.
Biose linear topologies which result from semimetrics and especially seminorms have many properties not
possessed by l.t.s. in general.
A seminormed linear to-
pology is always convex, and every locally convex topology
is the sup (to be defined) of a family of seminormed topologies .
Kelley and Namioka [2] define a function q on a
linear space E which is very similar to a seminorm.
By
definition q has the following properties:
(1)
q(x) ^ 0 for all x in E and q(0) = 0,
(2)
q(x + y) ^ q(x) + q(y) for all x, y in E,
(3)
q(ax) ^ q(x) for each x in E and each scalar a
such that |a| * 1, and
*.
.
•
.
8
(4) q(Vn) -• q as n -•«. for all x In E.
We will refer to such a function as a q-function>
but the notation !x! will be used instead of q(x).
If d(x, y) = q(x - y) = !x - y!, d is a semimetric
*
whose topology is linear. The neighborhoods of 0 are
circled and d is invariant, that is, d(x, y) = d(x + z, y -f z)
for all X, y, z in E.
•
Every linear topology is the sup of a family of linear
topologies determined by q-functions.
Completeness
In general a linear topology is not first countable, and nets must be used instead of sequences. A net
{x^: <* e A} is Cauchy iff for every neighborhood U of 0
there exists cc^ such that «^ /? > OCQ implies x^ - x^ e U.
If every Cauchy net in a l.t.s. is convergent, the space
is said to be complete.
If a linear topology is determined by a semimetric,
it is first countable and sequences may always be used in
place of nets. A sequence fx.j is Cauchy with respect to
the semimetric d iff for all e > 0 there exists N such
that n, m^-N implies d(Xy^, x^^) -< ^. A semimetric space
is complete iff every sequence converges which is Cauchy
in the semimetric.
The two types of completeness are not
always equivalent but when the semimetric is invariant
they are equivalent.
Category
A set A in a topological space is nowhere dense iff
the interior of its closure is void.
A topological space
is of the first category iff it can be expressed as a
countable union of nowhere dense sets.
It is £f the second
category iff it is not of the first category.
'Hieorem 2.9 (9.4, Baire's).
A complete semimetric
space is of the second category.
Theorem 2.10 (11.2).
A one-to-one continuous linear
map of a complete semimetric l.t.s. onto a Hausdorff l.t.s.
of the second category is necessarily a topological isomorphism .
Sups, infs, and lattices
Given a family 7* of topologies on some set, the
strongest topology weaker than each member of 3^ is denoted
by inf_3f and is the intersection of all of the topologies
in 7\
The weakest topology stronger than each member of 3^
is denoted by sup y^,
A subbase for sup^ is the collec-
tion of all sets open in at least one member of 7\
Now if J^ is a family of linear topologies on a
linear space, s u p ^ as defined above is linear and is not
only the weakest topology stronger than each member of 7^
but is also the weakest linear topology stronger than each
member of ^.
In general the intersection of a family of
10
•
k»
•
linear topologies fails to be linear (though it is always
a topology); and in general the topological inf of a family
of linear topologies (as defined above) is not the same as
the linear inf, which is defined to be the strongest linear
topology weaker than each member of the family.
Henceforth
the symbol "inf" will be imderstood to mean the linear inf.
Oheorem 2.11.
If ^ = l:?^: otcA} is a family of
linear topologies on a linear space E and 2^ is a local
base for jj, then a local base for sup jT is the collection
n
of all sets of the form H U^. where each U^ e ZC^..
1=1 '
Such a representation for a local base for i n f ^ is
still unknown in general, but Steiner [8] has constructed
a local base for the countable case.
Theorem 2.12.
Under the hypotheses of the previous
theorem a local base for inf ,5^ is {Z!Ui:U. e U±]
1=1
^
^ h a s n members.
when
When ^ is countably infinite, a local
base for i n f ^ is the collection of all sets of the form
U tU, + U^ + . . . + U_} where (uj is a sequence of sets
n=l
such that each U. e
21,.
Bie following theorem is easily verified.
Theorem 2.13.
Let
[X'
« « At be a family of
linear topologies on a linear space E and let
[5^:
oc € A\
be the family of relativizations to a linear subspace P.
11
Then sup IX^:
« € A| is the relativization to P of
sup (7^ : ec e A} .
'^'•*-
• • ' ^
It is a useful fact that the sup of all linear
topologies on an ^-dimensional space is convex.
Definition. A set A partially ordered by ^ is
called a lattice iff for each pair x, y e A there exist
unique elements x /\ y and x V y in A such that
(a) X V y $: X, y, and x V y « z for all z such that
z ^ X, y,
(b) X /\ y < X, y, and x A y ^ z for all z such that
z < X, y.
The following are lattices:
(1) the collection of all subspaces of a linear space;
P A E = P n E and F V E = F e E.
(2) the collection of all topologies on a set;
(3) the collection of all linear topologies on a
linear space; vT'-j^ V :7^ = sup(J^, S^l
as before but
J5_A J2 = inf{y^, T2I where "inf" means "linear inf."
A subset B of a lattice A is called a sublattice
iff it itself is a lattice under the partial ordering of
A iff it is closed under A and V .
12
Dimension
A highly important characteristic of any linear
space is- its dimension.
In fact two linear spaces are
isomorphic iff they are of the same dimension.
Definitions.
A set A in a linear space E is
linearly independent iff .ot-j^ = oc^ = . . . = oc = 0 whenn
ever Z^oc^ a^ = 0, each oc. is a scalar, and each a. c A .
1=1 ^
^
^
If A is linearly independent and every element of E can
be written as a finite linear combination of elements in
A, we say that A is a (Hamel) basis for E. A linear space
may have many bases but they are all of the same cardinality.
The dimension of E is the cardinality of any basis.
Definitions.
(l) F and G are complementary sub-
spaces of a linear space E iff G /I F = (Ol and G © F = E.
(2) If F is a subspace of a linear space E, the annihilator P*'' of F is the collection of all linear functionals
on E which vanish on P.
Verification of the following theorem is a straightforward application of the definition.
Theorem 2.l4.
space E,
If F and G are subspaces of a linear
i
(PHG)
i
i
= P-^© G
and
(p 0 G)^ = p^nc-^.
The following two facts are consequences of problem B, page 11, in LlJ.
(1) If P and G are complementary subspaces in E,
dim F = dim E/G and dim G = dim E/F.
(2) If F and G are any subspaces of E,
dim (P © G) + dim (FOG) = dim P + dim G.
By (1) we see that for any subspace P, the dimension
of its complementary subspaces is invariant; this dimension
is called the codimension of P.
Thus codim F = dim E/P.
By modifying slightly the discussion in [1, pp. 119, 120]
we see that E/F = P"^ whenever P is ,a subspace of E.
Now let P, G be subspaces of linear space E and
apply the above results.
codim P + codim G = dim E/F + dim E/G
= dim P-^ + dim G"*"
= dim(F"^ © G-^) + dim(F'^n G"^)
= dim(FnG)'^ + dim(F © G)*^
= codim(Fn G)' + codim(F © G).
Prom this formula we see that
Lemma 2.15.
The intersection of a finite collec-
tion of subspaces each of finite codimension is of finite
codimension.
Another useful fact is
Lemma 2.l6.
If P, K are subspaces of a linear space
E, codim of F H K in P ^ codim of K in E.
14
Proof: Note first that if A and B are linear independent sets in a linear space such that the two subspaces generated by them have,only
0
in common, then
A U B is linearly independent.
Now let Hj be a^ Hamel basis for PnK.
such that HjUAj^ = H^-, a Hamel basis for K.
H-t^Ap = Hp, a Hamel basis for P.
Let Aj^ be
Similarly let
Show card A^ * codim
of K in E. If the subspace generated by Ap had anything
except lOj in common with K, it would be PnK, and this is
impossible since Fn K and the subspace generated by A™ are
complementary subspaces of P.
Certainly Hj^ and A^ are
each linearly independent, so by the first remark above,
HrrUAp is linearly independent.
Thus there exists A such
that Ap £ A and Hj^UA is a Hamel basis for E. Clearly
card Ap ^ card A, and so
codim of K H F in P ^ codim of K in E.///
K
%
HI
Ap
subspace generated
by Ap
15
Miscellaneous
Ttie following inequalities will be useful.
(1)
If a, b are any comp'lex numbers and O^pssl, then
p
la + bl
P
p
^ lal + lb/ .
«
(2)
If a, b, X 5> 0, then
a
ax
.^^
,
and
(3)
a
ax
^ ^ — > ^r-T
iff x < l .
b + a
b -f ax
If a, b are any complex numbers,
/a + bl
lal
1 + la + bl ^ T T l a i
(4)
Ibl
1 + Ibj'
If a, b > 0,
a < b if f
^
< —
1 -f- a
1 + b
Lastly we point out the usefiil
Lemma 2.17.
cl(UnF) = clU whenever U is open in
a topological space and F is a dense subspace.
CHAPTER III
SPACES OP THE FORM l(p )
Definition
The l(p ) spaces studied by Simmons [7] are a generalization of the familiar iP spaces, 0 < p < l .
1^ consists
either of all real-valued sequences x = {x.{ such that
a0
^ lx.]^<oa
i=l ^
or of all complex-valued sequences with
the same property.
In either case if we let !x! =
P
2. |x I
i=l i
P
for each sequence in 1 , then I •! is a q-function.
1 with
the metric topology induced by ! •! is Hausdorff and complete.
To obtain the spaces studied by Simmons, consider
a sequence {p.j of real or complex numbers such that
0-<ip ^ 1 for all i = 1, 2, . . . Denote by l(p.) the col\x.\
such that
2" lx./ i<:«»,
^
1=1 ^
and for any element x = [x.\ of l(p.), define !x! = !E jx I
lection of all sequences
1
1
We will show that the !•! so defined is a q-function.
i = l -»-
De-
note by u(p.) the metric topology induced on l(p ) by ! •!,
and let (l(p.), u(p.)) represent
the collection l(Pj,) with
the topology u(p ). This space will be sho;m to be complete.
Because the metric induced by a q-function is invariant,
completeness with respect to the linear topology u(p^) is
16
equivalent to ccsnpleteness with respect to the metric induced by !.! (See page 8 ).
oo
Rote.
2^ will often be replaced by ]E, but the
1=1
bare summation sign will not be used for finite svmmiations
or for truncated infinite summations.
Pieorem 3.1.
!•! on l(p ) is a q-function and
(l(p ) , u(p )) is complete and Hausdorff.
Proof: We show !•! satisfies all the conditions in
the definition of a q-function.
!x! >0 for all x€ l(p.).
Clearly !0! = 0 and
Now given x and y,
X
»
Ix^Pi + l y ^ l ^ i ^ |x^ + y^|Pi for each i , so that
Zlx^l^i + J l y J P i ^ Z l x ^ + y j P i .
If / a / ^ 1 and x € l ( p ^ ) ,
lax! = riaXj-jPi = Zlal P i / x . / ^ 1 ^ Zlx^l ^i = !x!.
pick x e l ( p ^ ) .
Now
ix. = 2 : | ^ r ' = Zi^f'-l^i^^^'
m>n implies that !X! :^ !2£! .
•^
m
n
Clearly
So i t now suffices to pick
S>0 and show that there e x i s t s n sych that •^-"^ *^ex?
picking M big enough, we can make
^y
_
2" (1) ^Ix^l
i=M ^
oo
^
Z |x./^^-i£ for any value of n. Now pick n big enough
i=M
"•
2
so that (i)Pi|x,! ^ ^ < i , for each 1 = 1, 2, . . ., M - 1.
^n
1
2M
Bien
;•;?;•
/
.^'•-
18
So !•! is a q-function, and if !x! = 0, then x = 0, so
that u(p^) is Hausdorff.
We now show that (l(p^), u(p^)) is complete. Let
(X j ^ be a Cauchy sequence. Bien for each i = 1, 2, . . . ,
[x }
is Cauchy, because given i, there exists M^-o such
•1 N=l
that n, m > M implies
ix^ - :^! < / ^
Z/x^. x^l^i-/i
i
In
|x
i
miPi
- X
•JL
^Pi
1 ••• -^ ^
"^
J"
So let x^ —> y. for each i, and let y ={y^{
. First
i
1
i'i=l
we show that y e l(p.) and second we show that x —»• y in
Since {x } is Cauchy, it is bounded; suppose
^(PJ).
N
!x !^M for all N.
Let k be fixed and observe that
^
n P*
2!^ /x. I ^ ^ M for all n, so that by letting n - > ^ , we
1=1
^
k
P^
o b t a i n Z I y J ^«^M.
1=1 ^
So y £ l ( p ^ ) .
Now l e t k - > * ^
^
P*
t o get Z / y j ^ ^ M.
i=l ^
Pick €-5-0. There exists M~-0 such that if m, n^M,
IT
n
then 2r Ix"* - x^( ^<€. for any k.
1=1 i
^
Hold k and m fixed and
19
k :_
p^
l e t n—>oo, so that Z* 1^ - y J ^
i
^
-4* 1=1
...
P
so that ZT /x°^ - y.| ^ ^ e.
1=1 i
^
€ .
Now l e t k
Since the latter holds for
all m > M , x^--*- y.///'
Simmons has verified in detail the following properties of the spaces (l(Pj^), u(p^)).
I.
If 0<p^:^q^^l for all i, then
(1) l(p^) ^l(q^);
(2) the identity map (l(p^), u(p.))
"^ (l(q^)^ ^(q^)) is continuous;
(3) u(pjj^) on l(p^) is stronger than the relativization of u(q.) to l(p^); and
(4) l(p^) is dense in (l(qj^), \x{q^),
II.
If 0 < p ^ =^ q^^l for all i, then the following
three conditions are equivalent:
(1) u(p.) is the topology induced on l(pj^)
by u(q^);
(2) l(pj^) is closed in (l(qj^), u(q^)); and
(3) l(Pi) = l(qi).
III.
Suppose that 0-^r^^, Sj^ <i 1 for all i, and write
p. = min (r^, s^) and q. = max (r^, s^^). Then
l(Pi) = l(rj^)n l(si) and l(qi) = l(r^) Q l(sj^), the subspace of s generated by l(ri)U l(s^).
(That is, iCp^)
20
and l(qj^) are respectively the meet and Join of l(r,) and
l(s^) in the lattice of all linear subspaces of s. Recall
that s is the linear space
of all sequences.)
IV. Under the assumptions of property III the three
conditions l(rjL) S l(Sj,), l(pj.) = l(rjL), and l(s^) = l(q^)
are equivalent. Also l(r^) = l(s.) iff l(p^) = l(qjL).
Lattice of Linear Topologies
Bie above properties show rather clearly the relationships that exist between the l(p^) considered as
linear subspaces of s. Especially interesting is property
III.
One is prompted to ask if a relationship similar to
III exists between the linear topologies u(p.) and if
possibly they are a sublattice of some lattice of linear
topologies.
To answer the latter question we see that it
would be nice to somehow reduce all the u(pj^) to the same
set, and we are thus led to ask what sequences are common
to each of the spaces l(Pj^).
The following are an exten-
sion of Simmons' results.
Definition.
Let K*^denote the collection of all
sequences all but a finite number of whose entries are zero
Theorem 3.2. K"* is the intersection of all of the
l(p ) spaces.
Proof:
Clearly K^ is contained in each l(p^). Now
let {x.} be a sequence with infinitely many non-zero
k.V,
21
entries.
If IxJ ^ 1 / 4 or if |x f = 0, let p. = 1; and
— '*(««•-*
p.
if 0 -i|x^I <l/4, let Ix^l ^ = 1/2, that is, p^'log /x^l
= log 1/2 or Pj. =
[x^
log 1/2
log IXj^l
\
0 < p . ^ 1 and clearly
^ l(Pi).///
Definition.
Denote the relativization of u(pj^) to
K ^ b y w(pj^).
All of the linear topologies on K** form a complete
lattice and we will eventually show that all the linear
topologies of the fonn w(pj^) form'a sublattice but not a
complete sublattice.
First, however, let us look more
closely at the topologies u(p^), keeping in mind property
III.
Theorem 3-3.
Let 0 -^ r^, s^ ^ 1 and p^ = min(r^, s^)
©len u(p.) is the sup of the relativizations on l(p^) of
u(rj^) and u(s.).
Proof:
Denote the two relativizations by v(r^) and
v(si).
By property 1(3), ^{v±)
so that
U(PJL)
^ v(r^) and u(p^) 2 v(s^),
2 sup{v(ri), v(s^)j . Now let U be a neighPi
I
borhood of 0 with respect to u(p^). U 2 {x: Zlx^I
<^i
for some S ^ 0, and
22
V1 s:y^<
f .
,s.
{x: Zhjl ^-^finix:
ZU^\
^^i}
2' ' '
1
~ 2
•"
'•^
IJM-.:
Corollary 3.4.
Let 0 <^ r^^, s^ ^ 1 and p^^ = min(rj|^, Sj^).
Then w(pj,) = sup (w(r£), w(Sj^)}.
Proof:
See Theorem 2.13.
Theorem 3.5.
and p
Let 0 < r^, s^ ^ 1, q^ = max(rj^, s^),
= min(r^, s^). Then the relativization of u(qj^) to
l(pj^) = l(r^) O l(Sji^) is the linear inf of the relativizations of u(rj^) and u(sj^).
Proof:
We will mutually nest the two local bases
at the origin.
Denote by v(qi), v(rjL), and v(s^) the
various relativizations.
0 of the form Jx:
*•
2!lx.I
Given a v(q^)—neighborhood of
< ^„i where <f < 1,
1
Q"'
q
-^
= <5. = fc2 and show
choose 5«
^T.
r = Oes = --3.
(x:
Zlx^l ""^ ^ SJ + [x: ZlXil '^ < Sj
£ [x: r U i l ^ ^ ^
SJ.
Let X = y + z such that
r
Zly^l ^ ^ <r^ and ^Iz^l ^< (fg.
Then Z\x^\
23
* '
r
s
.q
Now given a neighborhood of 0 in sup | v ( r . ) ,
' r.
of the form {x:
Zlx^^l ^^
Sj[
^J^
(x:
choose S^ = min(cr, S ) ' ' Let x e [x:
q
^ r
v(s^)}
s'
••
Zl^±l
^
^Jy
2^fx.f ^ -i / f a n d
1
q'
write X = y + z where
rx^,
y. = f
•^. to,
if qi = r^
rx^, if q = s
and z. =V
otherwise
v o,
otherwise
Ohen Zlx^l"^^ = Zfyil""^ + Ziz^l^Vand Zlyil""^^ ^^ ^ /.
and
Zlzil^^^ ^q ^<^s-///
A proof virtually identical to the above will es-
tablish the following result.
Theorem 3.6. Let 0 -= r^^, s^ ^^ 1 and q^ = maxtr^, s^).
Then w(qj_) = inf fw(ri), w(Sj^)] .
The last theorem and Corollary 3-4 show that the
w(p.) topologies form a sublattice of the lattice of all
linear topologies on K**^.
Corollary 3.7. Let 0 < r^,
s^ ^ 1, p^^ = min(ri, s^),
and q^ =max(ri, s^). ^ e n the three conditions w(r^) = w(p^),
w(Sj^) S w(rj^), and w(qi) = w(s^) are equivalent. And
24
w(ri) = w(Sj^) iff w(pi) = w(qj.).
Proof:
We will use ^(q^)
= inf{w(r^), w(sj^)} and
w(Pj^) = sup {w(rj^) w(sj^)} . Prom the latter w(r^) = w(pj^)
iff w(s.) S w(rj^); from the former Mr{s^) s w(r.) iff
^(q^) = w(s.).
The last assertion is obvious.///
Lemma 3.8.
Then l(r.) j4
Let o'< r., Sj_ ^ 1.
l{s^)
Implies w(r^) j^ w(sj_), or equivalently w(rj^) = w(s;j^) implies l(rj_) = l(si).
Proof:
We prove the lattei* assertion.
Let
p^ = min(rj^, Sj^) and qj^ = max(r^, Sj^). By the previous
corollary, w(p^) = ^^{q^).
By property 1(1),
l(p^) e l(q^), so we now show l(qj.) ^ l(Pi).
q4
implies
{x€K*^:
Xlxj^l
x e l(q^)
f
<^ .
Zl^±l^^<S}
Now there i s a d > 0 such that
^
{XCK"":
0O
exists N > 0 such that
Z l^±l
i=N
ql
ZIX^I
< ^«
^^-^ i j and there
So for all M > 0,
N+M
qi
Z /xj
<d
i=N ^
N+M
p.
Z |xJ
< 1,
i=N
o©
p^
rix^r^^ 1
1=N
for all M > 0
?• ^\-. >'
TJ •
'«0
*
*
i=l
1
X€l(p^).///
Now consider a-mapping which associates each l(p^)
with the corresponding w(pj^).
It is certainly well defined
and the preceding lemma shows that it is one-to-one.
The
theorems Just proved show that meets are taken into Joins
and Joins into meets, so that
Theorem 3.9;
The lattice of linear subspaces of s
of the form l(pj^) is anti-isomorphic to the lattice of
linear topologies of the form w(p.) on K*^.
Recall from Theorem 3.2 that the intersection of all
of the l(Pj^) spaces is K**.
Now given a sequence {pj|^} such
that 0 < p. ^ 1 for each 1, consider the sequenceV--r-7—\
^
Ci^/Pi]i=l
which is in l(p.) but not in K*^
Thus K^^'is not a space
of the form l(p.), and we see that the sublattice of l(Pj|^)
spaces is not complete.
Hence the sublattice of linear
topologies of the form w(pj^) is also not complete.
CHAPTER IV
SPACES OP THE FORM L(p, /^)
Introduction
The last chapter was devoted to a generalization by
Simmons of the well known 1^ spaces.
The author of this
thesis has been able to generalize in an analogous way another well known space, iP.
If (X, s, zd) is a measure
space and p is a positive real constant, then X
is de-
fined to be the collection of all functions f measurable
n
'
on (X, s, /i) such that |fj^ is summable. Equality almost
everywhere is an equivalence relation in ^ ^ , and the collection of equivalence classes, denoted by LP, is a complete l.t.s. whose topology is determined by the following
q-function:
r
P
For background information on measure and Lebesgue
integration refer to Taylor [9], Chapters IV and V.
In
the following, several references are made to specific
theorems in this book.
We now prove several useful facts
not specifically mentioned in [9].
Lemmas on Measurable Functions
Lemma 4.1.
[0, 1] with
If f: [0, 1] —^ R* is measurable on
Lebesgue measure, and c is a positive real
26
27
'
•
•
'is"
constant, then c^ (defined by c^(x) = c^(^) for all
X € [0, 1]) is measurable.
Proof; We will show {x: c^(^)^b} is measurable
for all b e R . Since c^^^^ 1^ always positive it suffices
to consider only positive values of b.
{x: c^^^'^^o} = [o, i],
so now let b be finite. The theorem is clearly valid for
c = 1, so suppose c j4 1,
Then
{x: cf(^) ^ b} = {x: f(x)log c ^ log b}, and the
latter set is either
{x: f (x) ^ log b/log cl or {x = f (x) > log b/log c}
depending on whether o l or c <1.
Both are measurable
since f is measurable.///
Lemma 4.2.
If f, p are measurable on [0, 1] with
. Lebesgue measure such that 0 -^ p(x) and f (x) ^ 0 for all
X e [0, 1], then fP is measurable.
Proof:
Since f(x)P^^^ is always non-negative, it
suffices to let c ^ 0 and show that [x = f(x)P^^^ ^ cj is
measurable.
{x: f(x)P(^)^-'j = [0, 1] and
(x: f(x)P^"^^ ^ OJ = {x: f(x) = o}, which is measurable
since [0, 1] is a total measure space. If 0<c^«',
(x = f ( x ) P W ^ c/ = (x: f(x) =.c^/P(^)i
= jx: f(x) - cl/P(^) ^ of. 1/p is eveiyvrhere defined
and is thus measurable; by the previous lemma c '^P is
measurable and therefore (x: f(x) - c^/'Pi^)
measurable.///
< oils
28
Lemma 4.3.
If f(x) is measurable on [0, 1] such
that f ^ 0, then loir T(x) is also measurable.
Proof:
We must show {x: log f(x) ^ c! is mea-
surable for all c c R . Note that log «» = «> and log 0 = -oo.
Now {x: log f(x) ^ -oo}
= [0, 1] and
|x: log f(x) ^ -f oo} = jx: log f(x) = -f oo{
= tx: f(x) = +oo}, which is measurable.
Letting c be finite
we see that jx: log f (x) < cl = Ix: f (x) ^ e^}.///
Lemma 4.4.
Given that f, g are measurable on
[0, 1] and that f/g is defined a.e., let h(x) = f(x)/g(x)
whenever the quotient is defined ahd let h(x) be arbitrary
'.'.I -A'
elsewhere.
Then h is measurable.
Proof:
Since [0, 1] with Lebesgue measure is com-
plete, we can suppose h(x) s 0 on A = {x: f(x)/g(x) is
not defined!.
Let B = [0, IJ'^'A.
If c > 0,
(x: h(x) ^ c\ = A U {xeB: f(x)/g(x) ^ cl.
If c -« 0,
(x: h(x) i cf = {xeB:
f(x)/g(x) « c}.
It now suffices to show IxcB: f(x)/g(x) ^ c} is measurable for all C€R*.
{x€B: f(x)/g(x) 6ocJ
First observe that
= B
and
{ x c B : f ( x ) / g ( x ) ^ -coj = j x € B : f ( x ) / g ( x ) = - oo)
= {xe B: f ( x ) = +oo, - oo < g(x) < 0}
U { x € B : f ( x ) = -oo, 0 < g ( x ) < o^}
29
•
=
"
•
•
•
•
*
»
•
•
[{x€B: f ( x ) = + e o ]
n
(xcB:
U [{x€%rf(x) =«oe/
n
-oo<g(x) <: o}]
{xeB:
0<g(x)<ooi].
So l e t c be f i n i t e and observe t h a t
{ x € B : f ( x ) / g ( x ) ^ c} = { x € B : f ( x ) = -f-, - o < g ( x ) < o }
. U { x ^ B : f ( x ) = -oc, 0 < g ( x ) < o c j
U
{xeB: --<f(x)<-, f(x)/g(x) ^ cj
So it suffices to assume not only c but f(x) is finite and
to show that under these assumptions (x^B: f(x)/g(x) ^^ c/
is measurable.
But
f(x)/g(x) ^ c iff f(x) ^ cg(x),
if 0 < g ( x ) < «
f(x)/g(x) ^ c iff f(x) ^ cg(x),
if -oc<g(x)-^0
f(x)/g(x) « c iff c ^ 0
if g(x) = ±00.
,
Now consider the case that c ^ 0.
{x€B: f(x)/g(x) ^ c} = {xeB: f(x) < cg(x), 0<g(x)<«^/
U {xeB: f(x) :5 cg(x),-.oc»-cg(x) <0i
L/ [xeB: g(x) +-oj.
The case for c < 0 is the same except that
{x€B: g(x) = i *^i is replaced by the void set.
In either
case the proof is completed by observing that eg is measurable and that
{x€:B: f ( x )
< c g ( x ) , 0 < g(x)
U {x<£B: f ( x )
= {xeB:
U
^ cg(x),
<^]
--^-<:g(x)
^0}
(f - c g ) ( x ) ^ 0, 0 < g(x) -
[xeB:
(f ~ c g ) ( x ) 2^0, - *. < g(x)
^\
<^0}.///
m
•
•
•
•
•
:
.
.
.
•
.
30
Definition of L(p, MJ)
How let p be a measurable function on [0, 1] (henceforth understood to have Lebesgue measure) such that
0 < p(x) IS 1 for all x e [0, 1] and let -!^(p,^) denote
the collection of all* measurable functions f such that
ff/P is summable.
Clearly f e ;<f(p,^) implies \f\^
finite a.e., which implies f is finite a.e.
is
Recall that
"=a.e." defines an equivalence relation on the class of
measurable functions finite a.e. in such a way that the
equivalence classes form a linear space with the definitions
[f + g] = [f] + [g] and [cf] = c[f;|. Also if f € c^(p,/^,
g is measurable, and f = g a.e., then Ifj^ is summable and
|f/^ = Igl^ a.e.; hence g e X{p,^).
So we see that either
every member of a given equivalence class is in ^(p,^)
or no member of it is in ^(p,/^.
Now denote by L(p,/4
the collection of all equivalence classes which are subsets
of ;jf(p,y^.
We now show that L(p,/0 is a linear space by
showing
(a)
if f, g € ^{Vy/^>
(b)
if f € ;zr(p,^) and c is real, cf e ^(p,/z).
For (a) let f, g c .2f(p,//).
then f + g ^ *^(p,/^), and
Then If + pl^ is measurable
by Lemma 4.2, If/^ and lg|P are summable, and for all
X € [0, 1]
/f(x) + g(x)|
^ |f(x)l
so that If + g|P is summable.
•i,-'ii».-'''.t-
+ /g(x)l
31
««
For (b) let f e «if(p,^) and c be real. Ohen /cfi is
" *-r
P
measurable by Lemma 472, If/ is summable, and for all
X € [0, 1]
|cf(x)|
= Ic/
. |f(x)|
/cl /f(x)|
,
if fcl
^1
|f(x)/
,
if lei 4 1 .
Now that L(p, It) has been verified to be a linear
space, pick an equivalence class and a representative f out
r p
of it and define I[f]! = y If/ d/t. ^ If g = f a.e., then
p
Igl
p
= If I
r
a . e . and
well defined.
p
r* p
/ If 1 du, = I |gf c^^, so that ! . ! i s
We now v e r i f y that ! •! i s a q-function and
that the r e s u l t i n g semimetric topology i s Hausdorff (that
i s , metric).
(a) Ifl^O
for a l l f € - ^ ( p , / ^ ) ; !0! = 0 ; and !f! = 0
implies f = 0 a . e .
P(x)
(b) For a l l X e [0, 1 ] , | f ( x ) -f g ( x ) |
^ / f ( x ) | P W + |g(x)fPW.
Hence [if + gl d^ ^ jifl d^ +
jigi cl/t.
p(x)
(cJ
Let lal ^ 1.
Then l a f ( x ) |
J
(d)
p(x)
= la|
|f(x)|
p(x)
^/f(x)r'
p
f
p
lafl d/i ^ JIfl d/i.
Pick f e ;2f(p, /i) and l e t Aj, = ( x : p(x) ^ 1 | .
p(x)
32
A^ is measurable since p is measurable; for all n, A^ ^ ^n+1^
•-' >ihff
(J A
n=l
= [0, 1 ] ; lim niA ) = M[0,
n-^oo'
Hence lim
[9]).
.-*,
C
J
1] = 1; and lim JA;^)
n-^co'
"
= 0.
P
|f I d/t = 0 (see Theorem 5-5 III in Taylor
f
Pick N > 0 such that \
P
|f/ d/t < ^
and consider
if where m is some positive integer,
m
![if]! = 5
f*
l^flV
p
P
r
P
P
= 3^N( 5 ) '^' 'V' ^3^NJH^ ''' ^
m
I
P
If/ cU + I .
Aj^
Now by making m big enough, holding N fixed of course, the
last s\im can be made less than
^.
Henceforth whenever reference is made to L(p, ^)
it
will be understood that [0, 1] is the measure space, that
p is measurable on [0, 1 ] , and that for all x c [0, 1 ] ,
0 < p(x) ^ 1.
Denote by u(p) the topology on L(p, /i)
-:••
•
3 3
Induced by the q-function ! •!.
Sometimes we will use
(L(p, ytf), u(p)) to designate the linear space L(p, /^),
with the particular linear topology u(p); often we will
use merely L(p, ii) to represent the linear topological
space. Generally we will follow the standard practice of
using f rather than [f] to designate the equivalence class
in L(p, /d) which contains f e«?J(p, jtt).
When we refer to
Jp it will be understood that p is constant.
Theorem 4.5.
If {f^J is Cauchy in (L(p,^), u(p)),
then {f } is Cauchy in measure.
Proof: Pick e > 0 and let
En,m= i^- '^nW " ^ m W
^ ^ J'
We w i l l show that ju.{E^^^) -»• 0 as n, m - • <»«.
Let An,ni = i^^^.m'
KM
' ^m(^)' ^ H ^^^
Bn,m = {x- /fn(x) ' ^m(^)I < ^i'
1
i^n - ^ m ' V
^^^
^ L
'^n '
^m'V
= J
Ifn - ^J V - 5B,IJn- ^ml'^A
''Ti,m
SO /.(An,«) -
«^t^l&il. i£«.
0 and ^{B^^J _ * 0 as n,m —
n,m
and thus
34
/<\,m) " M^,nlj^
Pieorem 4.6.
A\,n^
- 0 as n, m -«..///
If {fj^J Is Cauchy in (L(p,^, u(p)),
then there exists f€L(p,^) such that f -• f.
Thus
(L(p,/e), u(p)) is complete.
Proof:
{f { is Cauchy in measure.
So by Theorem
5-6 II in [9] there exists f, finite everywhere and measurable, such that fj^ converges to f in measure and a subsequence {g^J of (f } such that g, -»> f a.e.
f e L(p,/^ and that
6>0.
\K
Now show that
\lf^ - f/ d/« -» 0 as n -• «». Pick
There exists M such that m,»n >-M implies
€ .
- 8m'V <
Let m be fixed at some value ^ M and
consider the sequence j|g - g^l {
.
"
™ n=l
Let
p
h(x) = lim inf Igj^(x) - gm(x)| . By Fatou»s lemma (Theorem
n
5-5 V ) , h is summable and
Jhd« < lim i n f J/g^ - gm' ^/^ ^ ^
n
But |gn - gml^—
P
Hence | f - gj^l
1^ - gm'^ ^'^' ^° ^ = 1^ - Sjj^l a . e .
ff
.P
i s summable and j / f - 6^1 ^
- ^'
Finally,
jif^ - fiV ^ IK ' Si'V + Ji^i - ^'V
35
where the last two integrals can both be made small by
manipulating n and 1.///
Relationships among the L(p, JLCI^S
For different -measurable functions p and q we generally get different spaces L(p, ju,) and L(q, /i)
different topologies u(p)- and u(q).
and
We now investigate the
relationships between these different spaces all of the
form (L(p, / * ) , u(p)).
Lemma 4.7.
If f c ^{v> / * ) ^ then f , f" e ^(p, /u).
Proof:
f"^, f" are measurable since f is measurable.
p
.P
P
P
If I is summable and (f*") , (f") as If I because
jfl = (f"*" + f") . (f"^)^ and (f")P are measurable and
therefore summable.///
Lemma 4.8.
If f £ L(p, /t),
{fj^l of simple functions such that
there exists a sequence
J'f - f^' ^
"^ ^
as n -^ oo ,
Proof:
Consider first the case when f ^ 0. Since
f is measurable, there exists a sequence If^l of simple
functions such that 0 « f^^ -^ f (Theorem 5-1 IX).
But
since f is simple, f is bounded and measurable. Hence
n
*^
n
f„ * I.(P, /*) and t - f^e
P
If _ f^l
^
0.
(TJieorem 5-5 1)
L(p, /t).
But f - f^ "* ° implies
By the monotone convergence theorem
•
•
. • •
^ How remove the restriction that f » 0.
and by the previous lemma f"^, f" *r ^(p, ^ ) .
3 6
f = f"*" - f'
so there are
sequences fg^^J and {h^^l of simple functions such that
J+
p
r
If - gn' <^ -^ 0 and
]|f" - h^^l ^
fn ~ Sji - hj^; fn ^^ simple.
f
-
f_
= f •*• -
p
Let
Then
+ hh - Sn
-
f"
-* 0.
= (f
' -•*• - En) + (^» - f")
If - fjjl * If"^ - g„l + If" - h^l
*
If - f„i ^ If' - g„r + If" - hj""
JIf - f„l^<v« ^ Jlf^ - gnl'^.v^ + Jif- - h ^ f V ,
and the last sum goes to 0 as n ^-* «».///
Lemma 4.9.
Let p be measurable on [0, 1] such
that p(x) >• 0 for all x G [0, 1]. Given ^ > 0, there
exists
€ > 0 and a measurable set A such that
/<(A) < S
and p(x) ^ €. for all x 4 A.
Proof:
For all n, {x: p(x) ^ 1/nl is measurable;
if m > n, {x: p(x) ^ 1/mJ S {x: p(x) ^ 1/n);
oo
i ={x: p(x) = ol = n {x: p(x) ^ 1/n}.
n=l
Hence ^(^) = 0 = lim
/^(x: p(x) ^ l/nf).
• •
n-*«»
N such that /*((x: p(x) « 1/N})
<£,
There exists
Let c ^ 1/N.///
. .
>
Theorem 4.10.
37
If p, q are measurable on [0, 1]
such that 0 -< p(t) ^ q(t) ^ 1 for all t e [0, 1], then
(1) L(p,>) a L(q, / ^ ) ;
,
(2) u(q) on L(q, /t) is finer than the relativization
of u(p) to L(q, /t);
.
(3) the identity map (L(q, ju), u(q)) -• (L(p, /t), u(p))
is continuous; and
(4) L(q, /JL) is dense in (L(p, /t), u(p)).
(1) f € L(q, /t) implies that f is measurq
P
q
able and that If I i s siammable. But /f| « If I V I
q
P
Proof:
and If I V 1 i s summable, so that \f\
(2)
Given S>0,
i s summable.
i t suffices to find )j > 0 such
that
{x€L(q,^) :
jlfl^^^ <^i 2 {x€L(q,^) :
jifl d/4<>j/.
Let A = {x: | f ( x ) | > l } and B = {x: /f(x)/ ^ l } ; then observe that i f f € L(q, ^ ) ,
f
IflV = f IflV + f IflV " f '^'V + J if'V.
'^[O, 1]
^
JA
•^B
^
*^A
•'B
Now by Lemma 4.9, there exists a measurable set C s B and a
constant k > 0 such that /i{C) < ^/4 and p(x) ^ k for all
X < B'^C.
Then
[ Ifl^dtt -c <r/4 and \
Ifl ^
Jc
•'B^C
y^ = S/S(S/k)
:s J
If I d/e. Now choose
B-'C
1/k
r
. Obviously »J -^ •J/4. If f ^ L(q,/<) such
•
that
J If! dfL < «, then
C
|f|dtt « f
38
|ff%
-^ 3'^^ 4^ ^7- i«t
D = {X€B-C
: |f(x)f > (<r/4)^'^l.
^ D ) -c / / 4 s i n c e otherwise
So
y jfl
'^D
^ < S/l\. and
^
\
D i s measurable and
Ifldit ^ S/kiS/k)^^
)
|f| d
*^B'>'C-'D
=w
ii «r/4.
>
Combining these obseirvations and choosing
f € L(q,^) so that
\|f|^^ < H, we finally get
^ »i + i / 4 +
c
^
]
lf\ dJi
S
k
P
Ifj dfc +
D
^ / / 4 + ^/4 + S/k +
k
\
Ifl du
*^B^C-'D
S/k.
(3)
is an obvious consequence of (2).
(4)
Let f e L(p,^).
By Lemma 4.8 there exists a
sequence of simple functions converging to f in the topology
u(p); L(q,/i) contains all simple functions because
^(q^/*) ^'^
^y part (1) of this theorem, and L
all bounded measurable functions.///
contains
39
Corollary 4.11.
The intersection of all of the
#f(p>>M) spaces is ;zr .
Proof;
note that X
Use part (l) of the previous theorem and
itself is of the form •5f(p,/t) where p(x) s 1
on [0, 1 ] . / / /
^eorem 4.12.
The union of all of the JJf(p,^) spaces
is the collection of all measurable functions which are finite a.e.
P
P
Proof: f G ;i^(p,yu) **• Ifl is sxammable => If I is
finite a.e.- «> f is finite a.e. Now suppose f is measurable and finite a.e. and let A = {x: f(x) = 3 ooj . Define
p(x) = 1 on A U
{x: If (x)| ^ 2j and on {x: If (x)/ > 2j in
such a way that |f(x)|P(^) s 2.
Note that /f(x)|P(^) = 2
iff p(x)log /f(x)| = log 2 iff p(x) = log 2/log|f(x)/.
Using Lemmas 4.3 and 4.4, it is clear that p is measurable
and that f c
:C{v>/^'///
Theorem 4.13.
If 0 < p(t) ^ q(t) *s 1 for all
t e [0, 1], the following three conditions are equivalent.
(1)
the topology induced on L(q,/t) by u(p) is u(q);
(2)
L(q,/^) is a closed subset of L(p,/^); and
(3)
L(q,/4) = L(p,/^.
Proof:
(1) =*> (2):
Since L(q,/^ is complete with
u(q), it is complete v/ith the relativization of u(p) and
so is a complete subset of (L(p,/t), u(p)).
because (L(p,/«), u(p)) is Hausdorff.
It is closed
40
!•
(2) => (3):
Observe that L(q,/4) is dense in
(3) = ^ ( 1 ) :
Note that the identity map
(^q*/*)^ ^(q)) "-• (I'(P^/*)> ^(p)) is one-to-one, onto,
linear, and continuous.
Theorem 4.l4.
Now apply Theorems 2.9 and 2.10.///
Let 0 < r(t), s(t) < 1 for all
t €. [0, 1] and define p = r/As, q = r\/s.
I'(q^/^) = L(r,/4) n
L(s,/*) and L(p,/4) = L(r,/^) © L(s,/t)
= the subspace generated by L(r,/*) U
Proof:
Then
L(s,yM).
Clearly L(q,/i) s L(r,>a) H L(s,/*) and
I'(P5/<) ~ ^(r^/^) ® L(s,/t).
Now let
r
s
f € L(r,/*) D L(s,/*), so that Ifl and Iff are summable.
But
Ifl^ ^ I f l ^ V ffl^
mable.
€ [ f | r + ffjS and so lf|^ i s sum-
Now l e t f e L(p,/d) so that | f l ^ i s s\ammable.
A = {t: p(t) = r(t)l
and B = {t: p ( t ) = s ( t ) ,
A and B are d i s j o i n t and A U B = [0, 1 ] .
f(t),-
<*' = [i
if t ^ A
if t € B
^ /f(t),
(t) = /
0
,
if t ^ B
if t € A.
Let
s ( t ) ?^ r ( t ) | .
Define
and
Then for all t e [0, 1], f(t) = h(t) + i(t).
But
Ih(t)|^ ^ |f(t)|P for all t £ [0, 1], so that h e L(r,/^.
And ll(t)|^ ^ |f(t)|P for all t is [0, 1], so that
i € L(s,/t).///
"Corollary 4.15.
The following three conditions
are equivalent:
(1) L(r,;^ s L ( s , ^ ;
'
(2) L(r,yu) = L(q,/.); and
(3) L(p,/<) = L(s,>).
Corollary 4.l6.
L(r,/^ = L(s,/t) iff
L(p,/4) = L(s,/4).
Corollary 4.17.
u(r) = u(s).
L(r,/«) = L(s,/t) implies that
Thus each linear space of the form L(r,^)
has a unique linear topology.
Proof:
•
L(r,/4) = L(s,/<) implies that L(p,/i) = L(q,/<).
Now apply Theorem 4.13 (3) ==^ (l).///
Corollary 4.18.
L(r,/t) s L(s,/<) implies that u(r)
is finer than the relativization of u(s) to L(r,/<).
Proof:
L(p,/i)= L(r,/t) s L(s,/<) = L(q,/<); now
apply the previous corollary and Theorem 4.10(2).///
Theorem 4.19.
Let 0 < r(t), s(t) -s 1 for all
t e [0, 1] and let q = r V s.
Then u(q) is the sup of
the relativizations of u(r) and u(s) to L(q,/^.
Proof:
u(q) is clearly finer than each relativi-
zation and thus it is finer than their sup.
Now let
U = {f: JIfl d/JL <S\ be a basic neighborhood of 0 in
(I'(q>/4)> u(q)).
^ e proof is finished if we show
-• 1 .-K-,
42
U 2 fx«L(q,/t):
J(f| d/i ^
J
S/2\
s
Ifl d^^<r/2j.
But this is clear since ff|*^ ^ \f\^V
Ifl^
^ |f/^ + \t\^
J//
Theorem 4.20. *Let 0 .c r(t), s(t) « 1 for all
t * [0, 1] and let p = r A s , q = r V s .
Then the rela-
tivization of u(p) to L(q,/A) is the (linear) inf of the
relativizations to L(q,/t) of u(r) and u(s).
Proof:
The relativization of u(p) is coarser than
either of the two other relativizations and is thus coarser
than their inf. Now let U be a neighborhood of 0 in this
inf.
U 2 {f € L(q, ): J l f / % < S^}
+ if e L(q, ): JIfl d^ ^
Choose S
Sj,
= min((f , S ) and let f e {f €L(q,
): Jifl d/e ^ / i
Define g(t) = f(t) if p(t) = r(t) j^ s(t) or if
p(t) = r(t) = s(t).
If p(t) = s(t) 7^ r(t), let g(t) = 0.
Similarly define h(t) = 0 if p(t) = r(t) / s(t) or if
p(t) = r(t) = s(t,).
If p(t) = s(t) 1^ r(t), let h(t) = 0.
Then for all t e [0, 1], f(t) = g(t) + h(t).
only to show that
g € {f€L(q,/.):
jIflV *^r^
h € {f€L(q,A):
p s
JIfl dfi < J^l.
and
It remains
*3
g was defined to be f on a certain measurable set
and 0 elsewhere, so g is measurable,
h is measurable for
the same reason.
q
|g(t)r ^ If(t)|^, \/te[0, 1]
implies g, h fi ^(q,^)
lh(t)|^ ^ If(t)|^, Vt^[0, 1]
Furthermore,
ig(t)r
| f ( t ) P , V t e [ 0 , 1]
ih(t)i^
| f ( t ) | P , t'teCO, 1]
implies
A Lattice of Linear Topologies
In order to understand the motivation and ultimate
goal of the next few theorems and corollaries, the reader
is advised to read the last paragraph of this chapter.
recall that L
Now
is precisely the collection of elements com-
mon to all of the spaces of the form L(p, ^ ) .
Denote by
w(p) the relativization of u(p) to L . Ihen the last proof
with a few obvious changes goes through to show
Theorem 4.21.
w(p) = inf{w(r), w(s)} if p = r A s
and 0 < r(t), s(t) « 1 for all t e. [0, 1].
The following is an obvious corollary to Corollary
4.18.
44
Corollary 4.22. L(r.^) £ L(s,/t) implies
•
#
-•-.%
w(r) s w(s).
Corollary 4.23. w(q) = sup{w(r), w(s)| if q = r V s
and 0 ^ r(t), s(t) ^ 1 for all t ^ [0, 1].
Proof: Use Theorem 4.19 and Theorem 2.13.///
Corollary 4.24. Let 0 -c r(t), s(t) ^ 1 for t c [0, 1]
and let p = r A s , q = r V s .
Then the following three con-
ditions are equivalent:
(1) w(r) 2 w(s);
(2) w(p) = w(s); and
(3) w(q).= w(r).
Also w(r) = w(s) iff w(p) = w(q).
Proof: Use Theorem 4.21 and Corollary 4.23.///
Theorem 4.25.
Let 0 <: r(t), s(t) -J 1 for all
t € [0, 1]. Then w(r) = w(s) implies L(r,/<) = L(s,/t).
Proof: Let p = r A s, q = r V s. Then w(p) = w(q)
and it suffices to show that L(p,/t) £ L(q,A).
yf\^^<^)
There exists ^ > 0 such that |f c L :
2 {f € L-*-: \\f\^dju,
K =
Pick / > 0 .
^ yfl . Now pick f e L(p,/d); let
r p
P
Jlf/ c^ and let E^ = {x: n - 1 ^ |f(x)/ ^ n},
E«, = {x: /f(x)| = ooj . Each E^ is measurable and
/«(E«, ) = 0.
Define g^ = f on E^^ and zero elsewhere. Each
*
oo
g^ is bounded and measurable and f = Z &n'
45
= \
j'f.L% = Z Llfl%
P
^
f
P
.l^'^E.^=
Z ^"[0,
J^ 1]IfXjV
%^
n=l
^n ^
n=iy[0, 1]
= Z
I
lg„l d/t.
n=l ^[0, 1] "
^
M+N r
p
There exists an M such that for all N, Z Jls^l d^ -^ ^.
M+N f
p
f
But Z JlgJ d/^ =
M+N
p
r
,M+N ,p
Z IgJ <V' = J
n=M *^ '^
^[0,1] n=M " ^
M+N
Since Z gn ^'^ bounded and measurable,
r
ZgnhV^-
•'[O, l]'n=M "' ^
.M+N iq
J[0, l]'n=M ^' ^
By the monotone convergence theorem,
J
. eo
.q
r
.M+N .q
ZgnM^ = N-*o*
li°^ •'[0,
K l]'n=M
JZgnhV^^^^'
[0, l]'n=M"' ^
So \^
jfl d/u ss /.
Since If I is bounded and thus sum-
n=M
. ,q
mable on each Ej^ for n = 1, 2, . . , M, Ifl is summable
on [0, 1] and f € L(q,/<), as was to be shown.///
Now denote by L the collection of all measurable
functions on [0, 1] which are finite a.e. All of the
linear subspaces of L form a lattice and Theorem 4.l4
shows that those linear subspaces of the form L(p,^) form
i4.':
46
a sublattice.
All of the linear topologies on L
form
a lattice and Theorem 4.21 along with Corollary 4.23
show that the linear topologies of the form w(p) form a
sublattice.
Now consider a mapping from the first sub-
lattice into the second which always associates L(p,/t)
with w(p).
By Corollary 4.17 this mapping is well defined;
it is clearly onto; by Theorem 4.25 it is one-to-one.
By
Theorem 4.l4, Theorem 4.21, and Corollary 4.23, it takes
sups into infs and infs into sups.
anti-isomorphic.
So the two lattices are
Formally we can state
»
Theorem 4.26.
L
The lattice of linear subspaces of
of the form L(p,/4) is anti-isomorphic to the lattice
of linear topologies on L^ of the form w(p) under a mapping which associates each L(p,/i.) with the corresponding
w(p).
CHAPTER V
PROPERTIES OF LINEAR TOPOLOGIES
Definitions
Klee [3] considers the following seven types of
linear topologies.
A linear topology for linear space
will be called weak iff every neighborhood of 0 contains
a subspace of finite codimension, convex iff every neighborhood of 0 contains a convex neighborhood of 0, and
nearly convex iff each point x of E'^clto} can be strongly
separated from 0 by a continuous linear functional
f(i.e., Vx 4 cKOJ, f(x) j^ 0).
It will be called nearly
exotic iff there are no non-trivial continuous linear functionals, and exotic iff no proper closed neighborhood of 0
contains a convex set radial at 0.
Klee follows Ives [4]
in calling a set X s E semiconvex iff X is starshaped
from 0 (i.e., [0, 1]X = X) and /(X + X) s X for some / > 0 .
A linear topology will be called semiconvex iff every
neighborhood of 0 contains a semiconvex neighborhood of 0,
and strongly exotic iff no proper closed neighborhood of
0 contains a semiconvex set which is radial at 0.
General Results
The theorems of this section are either stated or
alluded to in Klee [3]. Theorem 5-4 and Lemma 5-5 are
47
48
'«.
proved there.
Minor alterations have been made and proofs
have been supplied for the rest.
Theorem 5.1.
A.l.t.s. E is nearly convex iff for
every point x in E-cljO}, there exists a convex neighborhood W of 0 such that'x ^ W.
Proof:
only if:
Let x e. E-cl{0)and f be a con-
tinuous linear functional on E such that f(x) = a 7^ 0.
Then {x: /f(x)/ <• /a|} is a convex neighborhood of 0 in E
which does not contain x.
if:
Let X e E'^'Cl{0}and W be a convex neighborhood
of 0 such that x ^ V7. By the Separation Theorem 2.6 there
exists a non-trivial continuous linear fxinctional f on E
separating W and x.
Since W is radial at 0 and linear
functionals preserve lines, f(x) ^ 0.///
Theorem 5.2.
A l.t.s. E is nearly exotic iff there
is a proper convex neighborhood of 0.
Proof:
If f is a non-trivial continuous linear
functional, let W = f~-^[S]_(0)]. W is proper because f[E]
is a linear space other than the singleton (ol; W is open
because f is continuous; it is convex because an inverse
linear image of a convex set is convex.
Conversely if W
is a proper convex neighborhood of 0, let x ^ W and then
apply Theorem 2.6.///
Theorem 5.3.
The only exotic linear topology on an
^-dimensional linear space is the indiscrete topology.
49
Proof:
Now if X±s
The indiscrete topology is obviously exotic.
a linear topology other than the discrete
topology, let U be a proper neighborhood of 0. U H v, a
proper closed neighborhood of 0 and V H W, a neighborhood
of 0 in the strongest.linear topology. See Theorem 2.3.
By the remark after Theorem 2.13, this strongest linear
topology is convex and thus W a C, a convex set radial at
0. So U 2 C, and ,7"cannot be exotic.///
Theorem 5.4. A l.t.s. E is weak iff every neighborhood of 0 contains a set of the form
^ -1
*
n f4 ]-l, 1[ for f. <£ E .
1=1 ^
^
Proof:
The "if" part follows immediately from the
fact that a finite intersection of subspaces of finite
codimension is also of finite codimension (Lemma 2.15).
For the "only if" part, consider any arbitrary
neighborhood U of 0.
Let V be a closed neighborhood of
0 such that V + V s U, P a subspace of finite codimension
in V, and f the natural homomorphism of E onto the quotient space Q = E/clF.
The quotient space is Hausdorff
(See Theorem 2.2 (5)) and dim E/clF = codim clF ^ codim P.
So Q is finite dimensional and "Hausdorff and is thus
topologically isomorphic to a finite dimensional Euclidean
space. Any neighborhood of 0 in an n-dimensional Euclidean
AfcJU^ TECHNOLOGICAL COLLEft^
LUBBOCK. TEXAS
LIBRARY
VJP,''
50
" g7i[S2^(0)]
.-1
space contains a set^pf the form f]
where
each g. is a continuous linear functional,
f (V) is a
neighborhood of 0 in Q and thus there exist continuous
linear fiinctionals g^^,* . . . , g^^ on Q such that
f(V) 2 n gjL*" [Si(0)].
Let fjL = g^o f . fjL is clearly
linear and continuous, and
nfi^[Si(o)]= n^'^ogi\s^{o)]
1=1 ^
-^
^
1=1
= f"^[ ng:^[s.(o)]]
1=1 i
^
Sf"^[f(V)]
= UlA: A € <P(V)}
= tys y^'X f o r some x e V J
= fyi y — X € c l P f o r some x cV}
s V + clF s V + V € U . / / /
Lemma 5.5- In a l.t.s. E of the second category, a
closed set X radial at 0 has non-empty interior; if in
addition X is semiconvex, 0 e int X.
Proof:
Because E is radial at 0, E = (J nX; because
n=l
E is of the second category, not every nX can be nowhere
:i;;ili-,
dense.
Each of them is closed so some nX has an Interior
point p; then p/n is an interior point of X.
Now suppose
further that X is semiconvex and let Y = X H — X.
X, -X,
and therefore Y are closed and radial at 0, so by the first
part of the theorem, Y has an interior point q.
is semiconvex,
0 =
i'(X + X) s X for some / >• 0.
a^q + y{-q)
Because X
Then
€ /(IntY + IntY) s ^(int(Y + Y)) = int/(Y + Y)
£ int/(X + X) s intX.///
theorem 5.6.
Every weak topology is convex; every
convex topology is semiconvex and nearly convex.
Every
strongly exotic topology is exotic and every exotic topology
is nearly exotic.
Every nearly exotic topology of the second
category is exotic.
Every non-trivial strongly exotic topol-
ogy fails to be semiconvex.
Every non-trivial convex topol-
ogy fails to be nearly exotic.
Every non-trivial nearly
convex topology is not nearly exotic.
Proof:
borhood of 0.
(a)
Let E be a weak l.t.s. and U a neighn _i
By Theorem 5-4, U s n fi ] - i> ![• The
1=1 ^
latter set is convex because the inverse linear image of
a convex set is convex, and it is a neighborhood of 0
since each f. is continuous.
(b)
of 0.
Let E be a convex l.t.s. and U a neighborhood
U 2 A where A is a convex neighborhood of 0.
Clearly
A is starshaped from 0 and /(X + X) £ X when / = 1, so that
A is a semiconvex neighborhood of 0.
52
(c)
Let E be a convex l.t.s. and x e E-^cUo}.
Because c H O } = H {U: U is a neighborhood of O}, there
exists a neighborhood U of 0 such that x # U.
Hence there
exists a convex neighborhood W of 0 such that x ^ W.
The
conclusion then follows from Theorem 5.1.
(d)
Let E be a strongly -exotic l.t.s.
Suppose E
is not exotic, i.e., that there exists a closed proper
neighborhood U of 0 which contains a convex set A which
is radial at 0,
Then A is semiconvex, which contradicts
the fact that E is strongly exotic.
(e)
Let E be an exotic l.t.s. and suppose it is
not nearly exotic, i.e., that there exists a non-trivial
continuous linear functional f on E.
Denote by 82^(0)
the set of scalars whose absolute value is less than or
equal to 1, and let U = f"^[S^(0)].
Then U is a closed
proper neighborhood of 0 and is itself convex and radial
at 0.
(f)
category.
Let E be a nearly exotic l.t.s. of the second
Suppose that U is a proper closed neighborhood
of 0 and that U 2 K,'which is convex and radial at 0.
Then U 2 clK, which is also convex and radial at 0.
Lemma 5.5, clK is a neighborhood of 0.
By
So there exists
a proper convex neighborhood of 0, which according to
Theorem 5.2 contradicts the near exoticity of E.
(g)
Since the topology is not trivial, there exists
a least one proper closed neighborhood U of 0.
By the
53
«>
strong exoticity, U cannot contain a semiconvex neighborhood of 0.
-r-*^-^
(h) Use Theorem 2.7, letting A and B be two disjoint singletons.
(i)
If a linear topology is nearly convex, and
there exists a point x ^ cl{0}, then there exists a nontrivial continuous linear functional.///
k-convexity
Landsberg [5] has defined another property called
k-convexity which a linear topology may possess.
Except
for Lemma 5.10 and Theorem 5.12, the results of this section come from either [7] or [5]^ principally the latter.
Definition.
Let 0 < k « 1.
A set M in a linear
space is k-convex iff ax + by € M whenever x e M, y e M,
a 5» 0, b s» 0, and a^ + b^ = 1.
Definition.
A l.t.s. is said to be k-convex iff
every neighborhood of 0 contains a k-convex neighborhood
of 0.
If M is k-convex and 0 e M, pick x e M and let
lal r, 1.
Then |a|^ ^ 1 and lal^ + [(1 - la|^)lA]k ^ ^
imply ax + (1 - |a|^)^'^0 = ax € M.
0 to X is in M for all x e M.
Thus the line from
Hence if M is k-convex,
0 ^ M, a > 0, b > 0, and a^ + b^ -^ 1, we can assert that
ax + by fi M because
54
^TTT-TTrm^
(a + b )
*'^;\
^ ^k.lA^
(a + b ) '
= 1 and (ak + bk)lA 6 1.
Now suppose 0 « M, M is k-convex, and 0 < 1 « k.
Let
X « M, y « M, a > 0, b > 0, and a"^ + b^ = 1. Then
k
k
1
1
a + b ^ a
+ b = l , and by the preceding remarks
ax + by c M.
So M is 1-convex and we have proved the folr
lowing theorem.
Theorem 5.7. If a l.t.s. is k-convex and 0 .< 1 ^ k,
then it is also 1-convex.
But in general 1-convexity does not imply k-convexity
as we see in Theorems 6.5 and
6,6.
The following theorem affords an alternative definition of k-convexity in a form very useful for applications.
Theorem 5.8. A set M in a linear space is k-convex
n
iff Z^ aj.Xj^fiM whenever each x^ e M, each a^ 3» 0, and
Z a.
1=1 ^
= 1.
Proof:
The "if" part is obvious Just letting n = 2.
The "only if" part will be proved by induction on n.
Clearly the proposition holds for n = 2. Now suppose that
n-1
it holds for n - 1, i.e., that Z a^x. £ M whenever each
1=1
n-1 ^
XI « M, each a. ^ 0, and Z ai*" = 1. Pick y^ « M for
^
•
^
1=1
i = 1, 2, . . . , n and choose n non-negative scalars
i-
55
r
'
b , , . . . , b_ such,jthat Z"^ O k _= .1.
.Let
. . .B = ' Zb^
t-^ k.
"^
"
1=1 "
i=l ^
n-1
^
n-1
/ ir
Then Z 1/B(bi^)= Z (b./BVk) = i. By the induction
i=l
1=1 ^
hypothesis £ (ti/BlA)y^ e M. But (B^A)'^ + ^^
= i^ so
that B^^ 'r (bi/B^'^)yi •+ b^Yn = Z biy^ * M.///
i=l
1=1
Bie following definition of absolute k-convexity
appears in Simmons [7]. Landberg's definition of the
same concept is slightly different*.
Definition.
Let 0 < k « 1. A set M in a linear
space is absolutely k-convex iff ax + byfiM whenever
X € M, y ^ M, and lal^ + |bl^ ^ 1.
If M is absolutely k-convex, clearly 0 € M and M
is starshaped from 0.
In fact M must be circled.
The following theorem affords an alternate definition of absolute k-convexity in the form most useful for
applications.
Theorem 5.9.
n
A set M in a linear space is absolutely
k
zia.r^i.
1=1 ^ .
Proof:
The "if" part is obvious just letting n = 2.
56
The "only if" part will be proved by induction on n.
Clearly the proposition holds if n = 2. Now suppose that
n-1
it holds for n - 1, i.e., that Z a,.x^tfM whenever each
1=1
n-1
XifiM and Z
j^
la.i
^ 1. .Pick y. c M for 1 - 1, 2, . . .
n
and choose n scalars b-j^, b^, . . . , b^^ such that
ZIbil ^ ^ 1.
1=1
Let B = Z Ib.l ^ ^ 1. Then
i=l ^
n-1
'^"-'•1
j^
Z(l/B)lb,l
1=1
^
• ^ '
*
>
i/vik
= Z |bi/Bi/^| = 1. By the induction hypothesis
1=1
•
-
,
'z\bi/BlA)y ^ M. But I B ^ ^ I ^ +.|bJ^ = ZlbJ^ ^ 1,
1=1 ^
^
^
jSl ^
so that
B1A('2:
1=1
(b,/BlA)y ) + b y
^
Lemma 5.10.
in
= f by
^ M.///
^^-|_ 1 1
(l) The k-convex extension of a set
M (i.e., the smallest k-convex set containing M) is
n
i^ k
)
{x: X = >" a^x^, each x. € M, Z a . = 1 , each a,- > 0|.
1=1 i ^
"•
1=1 ^
^
(2) The k-convex extension of a circled set is circled,
(3) If a set M is k-convex and circled, it is absolutely k-convex.
Proof: (1) By the previous theorem anything of
n
the form Z aiXj^ will be in the k-convex extension, pron 1^
vided that each a. > 0, each x. e M, and Z a. = 1 .
i
i
1=1 i
Let c^ + d
= 1, c ^ 0, d 5= 0 and consider two sums
57
n
m
Z fti^i and Z '^±y±^t
JU^Jm
the above form.
Then
1—"JL
n
.
m
n
.
m
c Z aiXjL + d Z b^^y^ = Z (ca. )x. + Z (db. ) y i , and
i=l
1=1
i=l
•*• -^ 1=1 J- -^
Z (ca^)^ + Z ( d b , ) ^ = c'^Z a^^ + d ^ Z b , ^ = c^ + d^ = 1.
1=1
"^
1=1
^
1=1 ^
i=i i
(2)
M.
Let N be the k-convex extension of c i r c l e d s e t
n
If y « N, y = Z a..x^ where each x^ e M, each a^ » 0.
i—1
n
and Z a.
1=1 ^
=1.
n
by = Z (ba )x
1=1
^ 1
(3)
1
•*•
Let Ibl ^ 1 and show by e N.
n
= Z a . (bx.) e N because M is circled.
i=l ^
^
Let M be circled and k-convex.
and let lal^ + Ibl^ < 1.
Pick x, y € M
Show ax + by e M. If
A = ( lal^ + lbl^)^/^j[ lal/A)^ + (|b|/A)^ = 1.
a = fale^®
and b = jble^ so ax + by = lale"^®x + |bl e-^^y. Because
M is circled ei®x and ei^y e M; because M is k-convex,
(|al/A)e^®x + (lb|/A)e^^y £ M; finally because M is circled
and A ^ 1 , lale^^ + Ible^^ = ax + by e M.///
^eorem 5.11. A l.t.s. is k-convex iff every
neighborhood of 0 contains an absolutely k-convex neighborhood of 0.
Proof:
The "if" part is immediate since an absolutely
k-convex set is k-convex.
be a neighborhood of 0.
For the "only if" part let U
Then U 2 V, a k-convex neighbor-
hood of 0, and V 2 W, a circled neighborhood of 0.
If W^
58
denotes the k-convex extension of W, V 2 W
and W is
k-convex and circled by the second part of the preceding
lemma.
By part (1) of the same lemma, W
is absolutely
k-convex. U H W^ and W^ is certainly a neighborhood of
0.///
The notions of k-convexity and of semiconvexity are
related as follows;
Theorem 5.12.
If a l.t.s. E is k-convex for some
0 •< k-^ 1, then E is semiconvex.
Proof;
It suffices to show that an absolutely
k-convex set M is semiconvex. M is starshaped from 0 and
we wish to find y ^ 0 such that /{x + y) e M for all
X, yfiM.
iff ^^
It suffices that y^ + ^^ « 1, and /^ + /^ * 1
2'-^^.///
Preservation of Type
The following two theorems indicate which properties
are preserved by subspaces, sups, etc. Theorems 6.9 and
6.10 are of the same nature and could naturally be included
at this point. Observe the table on page 87. I will denote the set of indices {w, c, k, sc, nc, ne, e, se} which
represent the various properties under discussion. The
next theorem was stated and partially proved by Klee.
Theorem 5.I3.
For 1 ^ J «s 3, let Aj denote the set
of all 1 ^ 1 such that whenever F is a subspace of a
59
l.t.s. E
(1) then if £fta'Sproperty i, P has property i;
(2) and P is dense in E, then if E has property 1,
P has property i;
(3) and P is dense in E, then if P has property 1,
E has property 1.
Then A-j^ ={w, c, k, sc, ncj, A2 = A^^ U
{ne} = I'^le, se},
and A^ = I'^{nc}.
Proof:
(la) weak:
If U is a neighborhood of 0 in
P, U = P/1V where V is a neighborhood of 0 in E. V s K,
a subspace of finite codimension, and V/IP 2 KnF.
But
the codimension of K/OF in F is less than or equal to the
codimension of K in E by Lemma 2.l6.
(lb) k-convexlty and convexity: Let ^ be a local
base for E consisting of k-convex sets. Then
iFnVi
V e Zl] is a local base for P, and the intersection
of two k-convex sets (and therefore of a k-convex set and
a subspace) is always k-convex.
(Ic) semiconvex:
Ttie proof is like (b); you observe
that the intersection of two semiconvex sets is semiconvex.
(Id) nearly convex: Denote by J'and ^ the topology
of E and its relativization to P.
that X 4^-cliO].
Then pick x e F such
Since J?-cl{o} = F H
T-cllo],
X ^-J^-clio} and there exists a convex ^^neighborhood W
of 0 such that x ^ W.
borhood of 0.
Then x ^ W/IP, which is a if-neigh-
fei>;
60
(le) nearly exotic:
L^ on [0, 1] for 0-cpssl is
nearly exotic (See Theorem 5.6), but the subspace C of all
constant functions on [0, 1] fails to be nearly exotic.
If f c C, let 1(f) = \ f(t)dt.
*'0
functional.
I is clearly a linear
Now pick £ > 0 and let / = €
f
and f(t) s a, then !f! = \
implies l l ( f ) l
. If
\f\
< f
P
fi
P
P
P
Ifl d/t = \ |al dt = lal < iT =£ .
•'[0, 1]
Sofal ^ e
P
^
Jo
= j l adt{ ^ j |a|dt = /a/ -: e.
Qlius I is continuous at 0 and hence on the whole space C.
That the indices e and se are not in A^ will be
verified indirectly by showing that they are not in A^.
(2a) nearly exotic:
linear functional on P.
Suppose f is a continuous
Kien by Theorem 2.8 there exists
a continuous linear extension T on all of E and this is
impossible since E is nearly exotic.
(2b)
exotic and strongly exotic:
[32239] the existence of a dense
Klee asserts
^-dimensional subspace
J of S (to be defined on page 82), and one naturally supposes that the topology of the space he has in mind is not
indiscrete.
Then by Theorem 5.3 J is not exotic; by
Theorem 5.6 it is not strongly exotic.
But as we shall
see in Theorem 6.8, S is strongly exotic and exotic.
(3a) weak:
If U is a neighborhood of 0 in E,
U 2 V, a closed neighborhood of 0, and V H W, an open
61
neighborhood of 0. By Lemma 2.17 W = WTTF and by Theorem
n
_'-***-*
5 . 4 , PnW 2 /O f"*i] - 1, 1[ where each f. i s a continuous
1=1
^
linear functional on P.
Let g. be the continuous linear
extension of f. to E guaranteed by Theorem 2.8. It now
^i - 1
n _i
suffices to show O g^ ] -1, 1[ ^ H f^ ] - 1> 1[.
i=l ^
i=l ^
If X 6 p n n gT"^] - 1 , i [ , X € n fl i - 1 , i [ . so let
1=1 ^
i=l
X e (E^F)n n g^ ] - 1, 1[. There exists a net \x^] in
i=l
P such that x^ -• x.
Then for all 1,
gi(x^) -^ gi(x) e ] - 1, 1[. Now find a net [y^]
S
n fi ] - Ij 1[ such that y^ -* x.
1=1
open,
Since ] - 1, 1[ is
g. (x^) is eventually in ] - 1, 1[ for all i. Ihat
is, there exists ACQ such that ^ > a^ implies that
gjL(x^) e ] - 1, 1[ for all i; whence fi(x^) € ] - 1, 1[
for all i, and x^ e f'^]
- 1, 1[ for all i, and
X e f) fJ"^] - 1^ 1[. Now let the net (y^j be composed
*^ i=l
of all the x^»s such that a ^OL^,
{y^} is a subnet of
{x^} and thus y^ -> x,
(3b) k-convexity and convexity:
Let U be a neigh-
borhood of 0 in E. U 3 V, a closed neighborhood of 0, and
V s W, an open neighborhood of 0. By Lemma 2.17 ^ = WHF;
62
W/IP 2 K, a k-convex neighborhood of 0 in P.
But !C is a
neighborhood of 0 in E since K 5 o n P where 0 is an open
neighborhood of 0 in E and K 2 o n F = 0.
K i s still
k-convex and U 2 V e W 2 K.
(3c)
semiconvexity:
Same as 3(b) observing that
the closure.of a semiconvex set is semiconvex.
(3d)
nearly convex:
Klee [3:247] gives an example,
.jinvolving several concepts not introduced in this thesis,
of a nearly convex dense subspace of a space which is not
nearly convex.
^3e) -nearly exotic:
If f is a continuous linear
functional on E, its restriction to P is clearly linear
and continuous.
If f is non-trivial, so is the restriction
to P because a continuous function on a general topological
space is determined by its values on a dense subspace.
(3f)- exotic:
Suppose U is a proper closed neigh-
borhood of 0 in E such that U 2 K, convex and radial at 0.
U n P is a closed neighborhood of 0 in the relativized
topology for F, and U/OF 2 K/IF, which is convex and radial
at 0 in P.
Obviously a proper closed set cannot contain
a dense set, so U ^ F,
in P.
which means that U/IF is proper
This contradicts the exoticity of F.
(3g)
strongly exotic:
Same as (3f).///
Klee states the following theorem without proof.
• ' •
,'.>'*
63
Iheorem 5.l4.
If ^ i s a family of linear topologies
on a linear space E each having property 1 e I'^Jne},
then sup^has property 1 alsoPartial Proof:
(a) The exotic and strongly exotic
cases are intractable.
(b) weak:
Let U be a neighborhood of 0 in sup^
n
n
U 2 n ^,6. - n Pot where each U., e ZC^, a local base for
i=l i 1=1 i
and each ^ is a subspace of finite codimension. By
n
Lemma 2.15, f) "Pec^ is also of finite codimension.
1=1 ^
( c ) k-convex: Let U be a neighborhood of 0 i n
n
n
sup^.
U 2 n U<JC. S n Vflti where each V^ i s a k-convex
1=1 i
i=l ^
neighborhood of 0. In general any intersection of k-convex
n
sets is k-convex, so p ^OLA is a k-convex neighborhood of
1=1 ^
^
0 in s u p ^
(d)
semiconvexity:
l i k e (cJ.
(e)
nearly convex:
Let x e J^-cllO} for each
T^eS),' and show x e T-oliO]
where 7=
sup 7\
If U i s a
^neighborhood of 0, U = x + U», where U» i s a JTlneighborn
hood of 0; U» 2 n V^, and
1=1 ^
n
n
X + U» 2 X + n V^. = n (x + Voti) 2 (0}. Now taking the
1=1 ^
1=1
contrapositive we see that x ^ ^clfoj implies that there
exists an oc such that x e J5^-cl{o}.
Since ^ is nearly
64
convex there is a ^ -continuous linear functional f such
that f(x) ^ 0.
(f)
Clearly f is also »7^continuous.
Ihat the theorem is not valid for the nearly
exotic case is shown by the construction on pages 78-82.///
CHAPTER VI
l(p^), L(p,;r), AND S RELATIVE TO THE
PROPERTIES OP CHAPTER IV
Introduction
This chapter is primarily devoted to a detailed dis-^
cussion of certain l.t.s. which possess various combinations of the properties discussed in the previous chapter.
Our three main examples are l(Pjr), L(p,^), and S.
In [3]
Klee mentions only S and the special cases iP and L^, and
he gives no proofs.
Before looking at these examples we
make several observations.
The trivial or indiscrete
topology on any linear space has all of the properties,
i.e., it is weak, convex, k-convex for all k, semiconvex,
nearly convex, nearly exotic, exotic, and strongly exotic.
Any seminormed linear space is convex because Sg(0)
is easily sho^vn to be a convex set.
Every sphere around
0 in such a space contains clloj, which is a subspace.
And
if some sphere Sg(0) contains a subspace F, then necessarily
P SclfO} (If X e P andltxil / 0, then 11x11 = ^ < e
and llaxK > e
for some scalar a, which is impossible since ax e P.).
So
if E is a normed linear space, the only subspace in a
spherical neighborhood Sg(0) is {o} Itself.
If E is an
infinite dimensional normed linear space, it cannot be weak.
65
66
• (l(p^), u(p )) and (K-, w(pi))
We now turn to the question:
Which of the proper-
ties discussed in the last chapter are possessed by the
l(p^) spaces discussed in Chapter III?
Simmons [7] gives
a detailed proof of the following result.
Theorem 6.1.
(l(p^), u(pj|^)) is k-convex iff
i(Pi) 21''.
It is clear then that if {pj^} is bounded away from
0, (l(pjj^), u(pj^)) is semiconvex.
However if ipj^ -^ 0
as 1 -• *», (l(p.), u(pi)) fails to be semiconvex. To
prove this suppose (l(pj^), u(pi)) is semiconvex, whence
83^(0) 2 K 2 S^(0) and /(K + K) s K for some Y ^0.
show this is impossible.
We
First observe that if x-^, Xg,
. . . , x^ € K, then
P
n-2
n-1
n-1
yx-^ + y X2 + . . . + ^
Xj^_2 + ^
^-1+ ^
^n ^ ^•
Let Xy^ be that sequence with all zeros except for if ^^
in the k-th place.
Then for any n the following sequence
yj^ must be in K.
n
= (y^^/Pi, ^2/^/P2,
y^s^^'^s,
. . . , y^-l<f l/Pn-l, y ^ - l ^ V P n , 0,0,0, . . .)
n-1 4^
n-1 .
Then 'y ' ^ Z O'^i) = S Zy^^' ^'
1=1
i=l
large, /"z/^^i > 1.///
i=l
^f n is sufficiently
67
».
Ihat l(pj[) is not semiconvex when Ip^ —* 0 makes
it clear that semiconvexity is not always transferred to
finer topologies.
Theorem 6.1 shows that k-convexity is
not generally transferred to finer topologies.
The same
theorem shows that 1 -is the only one of these spaces
which is convex.
It is not weak however, because it is
not finite dimensional arid because it is normed (See Taylor
[10], p. 88). If S^(0) 2 P where F is a subspace other
than the singleton lo} and if x e E, then ax e S^(0) s P
for some scalar a and ax c F implies x e P; hence E = P,
a contradiction.
•
To show that (l(pj^), u(pj|^)) is nearly convex, recall that u(pi) is Hausdorff and let x = {x^} j^ 0,
x^ ^ 0.
Some
Keep n fixed and define a function f on l(Pi) by
f(y) = yn for all y e l(Pi).
Clearly f(x) f^ 0.
f(cy + dz) = cy^ + dz^ = cf(y) + df(z), so f is linear.
We now show that f is continuous at 0.
Pick e > 0 and let
T\
/ = €.
Then ly\ ^ S ^
Zly^l
n
^ <V - ^ |f(y)l^'' = l y j "" <
i=l
= e<^^i =^If(y)l ^ ^.
We have just constructed a non-trivial continuous
linear functional on (l(Pi), ^{v±))
is not nearly exotic.
so clearly this space
(Simmons [7] has an explicit de-
scription of the space of continuous linear functionals
on (l(Pi), u(pj^).) By Theorem 5-6
exotic or strongly exotic.
it also fails to be
In summary we have the following
68
Theorem 6.2.
Let 0 ^ p^ ^ 1 for all i = 1, 2, . . .
(l(Pi)* ^(Pi)) is nearly convex, not nearly exotic, not
exotic, and not strongly exotic. It is k-convex iff
k
1
1 ^ l(Pi). It is convex iff l{v±) = 1 , but in this case
it is not weak. If ip^^ --• 0 as 1 -• « , it fails to be semiconvex.
We would expect to get a very similar theorem for
the topologies w(pj_) on K**.
Theorem 6.3.
u(pi)).
(1(PJL),
First we prove
K** is dense but not closed in each
(K***, w(pi) is not,complete and not nearly
exotic.
Proof:
X = {x^
To show that K°* is dense, pick
€ l(Pj^) and note that for any e > 0 there exists
N such that
**
p.
Z Ix../ i < e. Define y such that y^ = x^ for
i=N+l ^
^
i = 1, 2, . . . , N and y^^ = 0 when 1 > N + 1.
Then y e K*"
^
Pi
*^
Pi
and !x - y! = Z Ix^ - y^l
= Z IxJ
^ e.
1=1
i=N+l
Since K**;^ l(Pi) and since K°* is dense in l(Pj^),
K^cannot be closed in l(Pi).
K**is not complete because
if it were it would be closed since (l(pj^), u(p^)) is
Hausdorff.
Now define f on K*^as follows: f(x) = x^, if
X = {x.} . f is easily seen to be linear. Pick 0<e <1
and let S = 6. Then \y\ < S
-=>Z\YA'^^
i=l
^ S -^ lyj^^ < S
69
*• .VPi
/f(y)/ = ly^l -^ ^
-^ / = e. f is a continuous
linear functional on TC which is not identically zero, so
(K*J w(p^)) is not nearly exotic.///
Corollary 6.4. w(p^). on K**is k-convex iff
(l(Pj^), u(p^)) is k-convex .
Proof:
Theorem 5.13-
The "only if" part follows immediately fromThe "if" part follows from the same theorem
and the fact that K** is dense in (l(p.), u(p.)).///
Reference to Theorem 5.13 yields the following summary.
Theorem 6.5.
Let 0 < p^, ^ 1 for all 1 = 1, 2, . . . .
(K**, w(pj^)) is nearly convex, not nearly exotic, not exotic,
and not strongly exotic. It is k-convex iff 1 s l(p.).
It is convex iff l{p^)
weak.
= 1 , but in this case it is not
If ipj^ —*' 0, it fails to be semiconvex.
Prom this theorem it is clear that convexity, semi-
convexity^ and k-convexity for any k do not generally carry
over to finer topologies. We also see that for any k, the
inf of two topologies may be k-convex although neither of
the original topologies is k-convex.
fk ,
Let Pi = f
^
lk/2,
if 1 is even
fk ,
and q^^ =<
if 1 is odd
lk/2,
if i is odd
if i is even
w(l/2) = inf {w(p ), w(qi)} but neither w(p^) nor w(q^)
are k-convex because 1 4 l(Pi) and 1 4 l(qj^).
70
(i-2Ar . l'^ t>utIl-^/^l * l(Pi), Kq^).
1=1
*
K** is ^-dimensional and the finest linear topology
on it is convex and therefore semiconvex by the remark
after Theorem 2.13.
Since K^admits a non-semiconvex
linear topology, we see that the sup of a family of linear
topologies may be semiconvex even though not every member
of the family is semiconvex.
The first two columns of the following table represent Theorems 6.2 and 6.5.
The remaining columns represent
similar theorems relative to the spaces still to be discussed.
(I'(P./^)^ u(p)) and (L , w(p))
As a second example we consider the L(p,^) spaces
discussed in Chapter IV.
Recall that p is a measurable
function on [0, 1] and that 0 ^ p(t) ^ 1 for all t e [0, 1].
In the following theorem certain restrictions are placed
on p.; the author suspects that the Iheorem remains true
when "if p is bounded away from 1" is replaced by "if
L(p,^) ^ L-^."
"if p is bounded away from O" can probably
be weakened somewhat.
There are probably some semiconvex
spaces for which p is not bounded away from 0.
If p gets
close enough to 0, L(p,/^) is possibly even strongly exotic.
Ohe proofs of some of the previous theorems dealing with
L(p,/t) and 1{VA) are substantially more complicated than
the corresponding proofs for L^ or 1^ precisely because
71
I
TABLE 1
•».•.... i , f c .
SflMfilARy OF THEOREMS
ft
PH
^—^
0
H
P4
*«-^
H
'*—'
1.
weak
—
.<-**
^•"^
H
P4
^
•\
ft
•
rs^
**—^
•
^
^ • ^
—
?J
^
.»
n
ftH
>—^
ft
^
>w
convex
^ ^ B
except
ll
3.
1,
5.
c-
semiconvex
nearly
convex
6.
nearly
exotic
7.
exotic
8.
\
k-convex
strongly
exotic
_
1^
—
—
iPi-*0 i P i - 0
0<P*p
+
0<P«p
+
+
+
—
^^
_
—
^^.
ft
ft
T
•^
+
^
.»
H
^
.^
ft
»^
—
—
\^
—
+
«»•
piP<l
••
P<P<1
y/^
- if p - if p
close
close
to 0
to 0
+
+
0<P^p
0<P^p
+
—
^
CO
_
except except
p«P<l
ll
ll
^
^•^
^•^
—
p^P<l
2.
^-^
4^>x
^ • w
p^P-^l
p<P<l
+
p<P<l
+
p«P<l
+
+
+
+
+
+
piP<l
p<P<l
—
+
•
—
0<P«p
0<P<p
_
72
p or {p^} is not required to be bounded away from 1 or 0.
Biese "bounded-away from" type restrictions reveal
to some degree the-nature of the properties that we are
discussing.
If we allow L(p,^ to get back too close to
normality, i.e. L , by letting 1 be a cluster point of
{p(t): t e [0, 1]}, then L(p,^) may fail to be exotic or
nearly exotic and it may become convex and/or nearly con»
vex.
On the other hand if we allow L(p,ytt) to depart too
far from L
by allowing 0 to be a cluster point of
{p(t): t € [0, 1]|, then L(p,y(^ fails even to be semiconvex and may become strongly exotic.
In the matter of
k-convexity we see the gradual retreat of L(p,yic) from
normalcy into pathology.
Back to the matter at hand, we have
Theorem 6,6,
If p is bounded away from 1, L(p,/*)
is nearly exotic, exotic, not convex, not nearly convex,
and not weak.
If L^ H L(p,/t), L(p,/^.) is k-convex; and
if L^ is k-convex, p JS k.
If p is bounded away from 0,
L(p,/t) is k-convex for some k; it is therefore semiconvex
and not strongly exotic.
However L(p,/4) may fail even to
be semiconvex if p is close enough to 0.
Comment.
The author strongly suspects that one
can prove L^ H L(p,/«) if and only if L(p,/^) is k-convex,
but he has been unable to do so.
Proof:
(a)
See Theorem 6.2.
L(p,/£) is of the second category so
by Theorem 5.6 the first assertion will be proved if we
73
show that L(p,^) is nearly exotic.
So let 0 < p(t) ^
PQ
^ 1 for all t e [0, 1] and
let W be a convex neighborhood of 0. W 2 Sj(0). By
Theorem 3.2 It
suffices to show that W is not proper.
Pick f e L(p,/£) such that f ?^ 0. Now choose n big enough
so that !f !/n-*-"Po ^ / and choose n points S^ such that
0 =
SQ
< Sj^ ^ Sg < . . . -i Sj^ = 1 and
I
|f| dxt = !f!/n.
This is possible because the
SAu of a summable function f is a
Lebesgue integral \
J[a,x]
continuous function of x (See Theorem 5-5 III).
fnf(t),
if Si_i ^ t ^ S.
(0
elsewhere.
,
Jg^'- = \
Igil 4^ = i
n \f\
^
J[0, 1]
^
J[Si_i, Si]
Now define
djL :6 n^°(!f!/n) </.
^
n
Therefore f = Z(l/n)g_. ^ W as was to be shown.
1=1
k
(b) Let L(p,/^) s L . Define q(t) = max(p(t), k)
for a l l t e [0, 1 ] . - By Corollary 4.15, L^ 2 L(p,/^) = L(q,/^)
and by C o r o l l a r y 4.17 i t s u f f i c e s t o show (L(q,/^), u(q)) i s
k-convex.
We do t h i s by showing
q
Ifl d/i < S] i s k-convex f o r some
k
k
arbitrary S. Let f, g € S^(0), a 2= 0, b 3^ 0, and a + b = 1.
I
Then
74
Jlaf + bgl^^4et ^
J/afI d/i +
Jibgl d^
a Ifl (^ + ^b Igl
^
« a*" Jlf/V + b" Jigl V
< a V + b'V =<r
P
Now suppose L i s k-convex and t h a t k > p .
Then
S^(0) 2 K 2 S^(0) where K i s absolutely k-convex.
Pick
n > 1/^ and define
fn^/PcT^/P,
if t e (i/n, i+l/n]
gi(t) =f
Define g^(0) = n^/P^T^/P.
i0
,
elsewhere
Ig^! = l / n ( n / ) = S. Now l e t
a^s »_iy_ so that Z ai = n(l/n) = 1. Thus
ni/^
1=1
n
Z aj[gi e Si(0).
1=1
n
But Z a.g. is constant on [0, 1] with
i=l ^
the value n^cT"^/^ so that ! Za.g.! = (n/n^/^O^ = (n^""P/^)^
nVq
1=1 ^ ^
All of the above is valid whenever n >- 1//, and as n —> «?
so does n
P/^^, so we have a contradiction.
(c) Ihe third assertion is clear from (b) and
Theorems 4.10 (l), 5.6, and 5-12.
(d) It only remains to find p such that L(p,/4) is
not semiconvex. Let p be such that /^(E^^) > 0 for all n
and /<(E^)
does not converge to 0 as n -> °«^ where
Ey^ = (t € [0, 1]: 1/n+l < p(t) ^ 1/n}.
Clearly the \ ' s
75
eo
are disjoint, [0, 1] '^ U \y
--'•>.-* n=l
and Z A \ )
n=l
= 1.
Tb
see that such a p actually exists let E = (1/n+l, 1/n]
for all n > 2 and let E;^ = (V2> 1] Ulol.
" n " 'irn:
^ ^
Ihen
" n(n + 1) '
and
1/n+l
l - - ^ — ]
n(n + 1)
1
[n(n + l)]i/n+l
which converges downward to 1 as n -• «» as can readily be
verified by the techniques of elementary calculus. Now
define p(t) = 1 if t e E_,.
n
"
Suppose now that L(p,/0 is semiconvex.
S3L(0) H K 5 S^(0)
are now fixed.
and
^(K + K) s K; 0 -= /' ^ 1; i' and
Define C^^ = max(l, {f//tiB^)
).
f
Define
if t e Ej^
Then
if t e E^
r
p
r ,n
p
r
i/"^
P
n,P
f ,^ >l/n.
(c ) d/4 ^«'E„
\ (r
Jrn.n _ (^ = ''E[0,1] "
^
-^En
/
,
.^(E^),
_ i/n
1/n ,
,
JEn
if c ^ = (//A(E„))".
if C„ = 1
eo
Since Z/(E ) = 1, we can find N such that n > N implies
n=l
that C^ = (///i(En))'^. We will deal only with g^ such that
76
n 5S" N. All of these gj^»s are in Sj-(O). For simplicity
consider the g^^'s to have been renumbered beginning at
some integer greater than n. . Now for all m
2
n-1
n-1
f = ^g1 + ^ g^
2 + . . ^ + >' gm-1 + ^ g«
m ^ S1 (0).
But
5[0 J^' ^ %?1 IE|^ ^nl ^ - J,^ ^"V*
p
m-1 r
n
p
r
m-1
P
n=lX
m-1 r n 1 /w • P
^ Z \ (^ ) S ^ ^
n=l^E^
because Y ^ 1
m-1 r
1/n+l
^ Z I ^<^n
^ ^
n=l ^'En
because C^^ as 1
m-1
1/n+l
n=l
m-1
n/n+1
^ yZ^(///^(E^))
m-1 n/n+1
A(En)
1-n/n+1
n=l
m-1
^yz/A\)
f- *• *. Ii
*
1/n+l
n=l
m=l
1/n+l
- ySZ M\)
-"~ as n -- o^.
n=l
This is a contradiction and the proof is finished.///
77
_
.
1
If we recall from Theorem 4.10 (4) that L is a
dense subspace of any"L(p,^, we get the following
corollary of Theorems 5.6, 5.13, and
Corollary 6.7. w(p) on L
6.6.
is respectively weak,
semiconvex, nearly exotic, exotic, or k-convex iff (L(p,yit),
u(p)J has the same property.
In addition if p is boimded
«
away from 0, w(p) is not strongly exotic; and if p is
bounded away from 1, w(p) is not nearly convex.
Prom this corollary and Theorem 6.6 it is clear
that convexity, k-convexity, and semiconvexity are not
always transferred to coarser topologies. The same is
true for near convexity since (L , u(l)) is convex and
thus nearly convex. We see also that the sup of two
linear topologies may be k-convex although neither of the
original topologies is k-convex. Let
fk ,
p(t) = <
lk/2,
^
'
, , ^
a(t) = 1
if t e [0,1/2]
and
if
- - t ^ (1/2, 1]
if t € [0, 1/2]
• One can easily show that
if t € (1/2, 1]
M P W 4 ^
and L{q,/i)$
k-convex.
But sup{w(p), w(q)l = w(k), which is certainly
L^ so that w(p) and w(q) are not
k-convex.
In order to get an example of two nearly exotic
spaces whose sup is not nearly exotic, one is tempted to
let k = 1 in this last example.
L"^ is certainly not
78
nearly exotic and probably L(p,/^ and L ( q , ^ are. It
turns out however, rather surprisingly, that L(p,yi^ and
L(q,/£) are also not nearly exotic, because there is a
proper convex neighborhood of 0.
Consider L(q,/<). Ob-
viously what we have done for [0, 1] could be done for any
.interval [a, b ] . L-^ on [1/2, 1] is not nearly exotic and
so there is a proper convex neighborhood U of 0.
Let V be
the collection of functions in L(q, ^it) which agree on
[1/2, 1] with some member of U.
because U is.
V is proper, and convex
Suppose U contains the sphere S (O) on
[1/2, 1 ] , Then V contains the sphere S^(0) on [0, 1] because
J
, ,P
Ifl cLit
[0,1]
^
p
ho, 1/2]Ifl
\
d;t +
p
\
Ifl (^ <: /
*^[1/2,1]
1/2
Ifl dm + \
J[0,l/2]
J
r
^
\fl6fi
<• S
J[l/2,1]
Iflc^ -«:/.
[1/2.1]
One could probably prove that if /i\t e [0, 1 ] : p(t) = l} > 0,
then L ( p , ^ is not nearly convex.
In any case we still need an example of two nearly
exotic topologies whose sup fails to be nearly exotic.
(See the remarks after Theorem 5.l4.)
tion of a construction by Klee.
We use a modifica-
Letting L be an ^-dimen-
sional linear space we construct two nearly exotic linear
79
«»
topologies T
and ^
on L x R such that sup(jr, 71) is
not nearly exotic. R is the real field.
ProofI
Since there exists a nearly exotic Haus-
dorff ^-dimensional l.t.s. (see remarks in part (2b) of
Theorem 5.13)5 L can be given a nearly exotic Hausdorff
topology 7^.
Let Zlhe a local base for 71
Let 71 be the
linear topology on L x R v/hose local base U-. is the
collection of all sets of the fomi U x R with V e tC,
Let ^
^
be the linear topology on L x R whose local base
is the collection of all sets of the form
U x{of + R(y, 1) vfith \J e Zl and y a fixed member of L.
Note that R(y, l) means all real multiples of the ordered
pair (y, l).
(y, 1)
R(y, 1)
80
t(^ Is a special case of ZL with y = 0, so it suffices
to verify that tl^ i*s"in fact a local base. We use Theorem
2.1.
(1)
[U^ x{0{ + R(y, 1)] n [Ug x{0? + R(y, 1)]
2 [U^ xfol n.Ug X {0}] +
= (U^ n Ug) X lo) +
[R(y, 1) nR(y, 1)]
R(y, 1)
s U3 x(0} + R(y, 1).
(2) Given U x{0) + R(y, 1), pick V + V £ U. Tnen
V x{0} + R(y, 1) + V xlO} + R(y, 1)
s V x{6{ + V x(0i
+
= (V + V) X {0} +
s u X 0
R(y, 1)
R(y, 1)
+ R(y, 1).
(3) Given U x{0} + R(y, 1), choose V circled such
that V s U.
Let |a| ^ 1. Then
a[V xlOl + R(y, 1)] = a[V xlOi] + aR(y, 1)
-aV xlOl + R(y, 1) s u x{0| + R(y, 1).
(4) Given (x, r) e L x R and U.x{0j + R(y, 1). Note
that (x, r) = (x - ry, 0) + (ry, r) and that (ry, r) €R(y, 1)
There exists N such that a 5= N implies x - ry e aU. So
(x - ry, 0 ) ^ aU x (Oi = a[U x (0}]
and
(x, r) = (x - ry, 0) + (ry, r) ^ a[U x(Ol] + R(y, l)
= a[U xfOl + R(y, 1)].
We now show that ^^^ is nearly exotic.
proof with y = 0 shows that ^
The same
is also nearly exotic.
fe
81
Suppose f is a linear functional such that f(y, 1) 7^ 0.
Every neighborhood of"0 contains R(y, 1) so f is \inbounded
on every neighborhood of 0 and hence not continuous. So
suppose f(y, 1) = 0 and that f is continuous and linear.
Olien f/(L x{0}) is continuous and linear on L x {0} with
the relativized topology J?. But
U x(0| + R(y, 1) n
L X {0} = U X {Oj
whenever U e U, so that L x {0} with jB is topologically
isomorphic to L with the nearly exotic topology 7^.
Ihus
f is identically zero on L x (Of. Now if (x, r) e L x R,
then
(x, r) = (x - ry, 0) + (ry, r)
and
f(x, r) = 0 + rf(y, 1) = 0 + 0.
Thus f is identically zero on L x R.
We now show that sup {7^, J7^) is not nearly exotic
by constructing a continuous linear functional f.
Let f
be 0 on L X {Oj and let f(y, 1) = 1. f is clearly linear;
to show that f is continuous we find a neighborhood of 0
on which f is bounded (See Theorem 2.5).
Pick a circled
neighborhood V7 of 0 in (L, *7^ such that y ^ W.
.7'is Hausdorff.
Then there exists a circled neighborhood
U of 0 such that U - U s W .
U x R O U
a neighborhood of 0 in sup(^, 7^),
D
Recall that
xiOl + R(y, l) is
Let z e U x R
U xfO} + R(y, 1). z = (u, r) and z = (u», 0) + (ay, a)
= (u» + ay, a), so r = a and u = u» + ry. Then Ir/ < 1
82
since ry = u - u' € w, W is circled, and y * W.
f is
bounded on U x R A - ^ J x{ol + R(y, i) because f(z) = f(u, 0 )
+ f(0, r) = 0 + r.///
The Space S
As a final example consider the collection of all
measurable functions on [0, 1] with Lebesgue measure which
are finite a.e.; recall that "=a.e." is an equivalence
relation and that the equivalence classes form a linear
space.
Pick a member f out of the equivalence class [f]
\
and let ![f]! =
[£!
d/t. Clearly f = g a.e.
*^[0,1] 1 + Ifl
\
* . .
J 1 + Ifl
^
=
\
!^!
da, so !•! is well defined.
J 1 + Igl
With the help of Inequalities (2) and (3) on page 15, !•!
is readily seen to be a q-function which generates a
Hausdorff topology.
This linear space of equivalence
classes V7ith the Hausdorff linear topology generated by
!•! will be called S. As before we will often write f to
mean the equivalence class [f].
It is straightforward to show that convergence in
the metric topology is equivalent to convergence in measure.
Let fji —* f in measure and pick e > 0 .
that n > N implies ju{x'. \f^M
There exists N such
- f(x)l ^ ^/2]
using inequality (4) on page 15>
^ e/2. Then
83
I
l^n - fI
1+Ifn-fl
-^--
""
1 +
e/2 + e/2.
Now suppose r
'^" " ^'
(V^ -* 0 as n -* CO . Pick
e > 0 and / ^ 0, and let E^ = {x: |fj^(x) - f(x)| ^
ej.
There exists N such that n :^ N implies
I
II
J 1 + Ifn - fi
/K\)
d^ <
€<f
1 +€
,
Then n s- N implies
^ ^ because otherwise by inequality (4) on page 15
and by Theorem 5-21,
[
^^^ ' ^'
dM. ^ ( §
)/
J 1 + Ifn - f I
1 + €
for some
n 5- N.
Theorem 5-6 II now shows that S is complete.
By
Baires* Theorem 2.9, S is of the second category.
Theorem 6.8.
The complete metrizable l.t.s. S
defined above is strongly exotic.
Therefore it is exotic
and nearly exotic; it fails to be weak, convex, semiconvex,
and nearly convex; for all k, it fails to be k-convex.
Proof:
To show that S is strongly exotic suppose
W is a proper closed neighborhood of 0 such that W 2 K,
semiconvex and radial at 0.
W s clK, closed, semiconvex.
^
84
and radial at 0.
By Theorem 5.5, 0 e intK. So K is a
proper closed semfcdniBrex neighborhood of 0. We show that
this is impossible.
Let f e s and suppose ^ K + K) s K.
Becall that Yx^ + Y \
+ ; • • + ^'^''^^i + ^'^'\
^ K
Whenever x^, . . . , x^ € K because of the semiconvexity.
Suppose K 2 s^(0).
Pick n > l/S and for i = 1, 2, . . ., n - 1
define
/.x
((V/? )^(t).
^ > gl(t) = <
i 0
,
T . g^(t)
f.^ = jf(l//?''"^)^(*)^
I^t
V0
,
if i-l/n ^ t < i/n
elsewhere
if n-l/n < t < 1
elsewhere.
n-1 4
n-1
Clearly each g^ e S^(0) and f = Z r"g^ + Z^" g^^ « K,
which means that K cannot be proper.
The theorem's second assertion follows immediately
from Theorem 5.6.///
Two General Theorems, Table o'f Properties
Ihe nature of the following two theorems would place
them in Chapter V but they are included here since the
proofs depend on the examples in this chapter.
Theorem 6.9.
If a linear topology is weak, nearly
exotic, exotic, or strongly exotic, then any coarser linear
topology has the same property.
These properties do not
necessarily carry over to finer topologies. Convexity,
M
' 85-
k-convexity, semiconvexity and near convexity do not
necessarily transfer to either finer linear topologies
or to coarser ones.
Proof;
The first assertion is obvious. To show
that some property mentioned above is not necessarily
transferred to a finer linear topology simply note that
there is a l.t.s. without that property and that the indiscrete topology on the same linear space is a coarser
linear topology with the property. For the other half of
the last assertion see the remarks immediately following
Corollary 6.7.///
Theorem 6.10.
If ^ i s a family of linear topologies
all of which are respectively weak, nearly exotic, exotic,
or strongly exotic, then inf .?^ has the same property. If
^ i s a countable family of linear topologies each of which
is respectively k-convex, convex, or semiconvex, then inf .?^
has the same property.
Proof:
The first assertion follows from Theorem
6.9.
When J^ is countable a local base for inf ^ is the
collection of all sets of the form U{Ui + . . . + U^l
n=l
where each U^ ^ tt^,
2.12.
a local base for j : e 5^. See Theorem
If each J^ is k-convex, then each ^^ can be picked
to consist solely of k-convex neighborhoods of 0, and
each 0 {U, + . . . + U^} is k-convex because
n=l ^
:-!}•:
^- '
•
•••
8 6
(a) Any finite sum of k-convex sets is k-convex.
(b)
If each se-t-in a family ^ i s k-convex and the
union of any two members of 7^ is contained in a third member, then the union over all members of .7^ is k-convex.
Verification of (a) and (b) involves only a straightforward
application of the definition.
(a) and (b) remain true if "k-convex" is replaced
by "convex" or "semiconvex."
In fact to get a proof for
the "convex" and "semiconvex" cases of the theorem, simply
replace in the above proof every occurrence of "k-convex"
by "convex" or "semiconvex."///
Obsei^ve that the finest linear topology on a linear
space is always nearly convex since every linear functional
is continuous.
Since there does exist a non-nearly convex
l.t.s., we see that the sup of a family of linear topologies
not all of which are nearly convex may still be nearly convex.
The following table is a summary of this last observation, the preceding two theorems. Theorem 5.14, and
several remarks made in this chapter.
It is not known
whether the inf of a family of nearly convex linear topologies is nearly convex.
In the table "--" means that
there exists a counter example which precludes any positive
assertion.
The second coliomn for sup indicates when every
member of a family^ has a certain property Just because
87
sup.?^' has It.
The positive results in this column are
obvious from the first column.
TABLE 2
TRANSFER OF KLEE'S PROPERTIES
fineness
sup
(a)
inf
(b)
1. weak
2.
convex
Ctbl.
3.
k-convex
Ctbl.
4.
semiconvex
Ctbl.
5. nearly convex
6.
nearly exotic
C
7.
exotic
C
8.
strongly exotic
c
88
LIST OP REFERENCES
•> «k# .-^
[1] J. L. Kelley, General Topology, D. Van Nostrand
(1955), Princeton.
[2] J. L. Kelley, I. Namioka, et al.. Linear Topological
Spaces, D. Van Nostrand (I963), Princeton.
[3] V. Klee, Exotic Topologies for Linear Spaces,
General Topology and its Relations to Modem Analysis and Algebra, Academic Press (I96l),-New
York, 238-249.
[4] R. T. Ives, Semi-convexity and locally bounded spaces,
Ph.D. Thesis, 1957, University of Washington,
Seattle.
[5] M. Landsberg, Lineare Topologische Raibie, die nicht
lokalkonvex sind. Math. Zeitsch. 55 (195^;,
104-112.
[6] S. Mazur and W. Orlicz, Sur les espaces metrlques
lineaires. I. Studia Math. 10 (194t5}, lti4-208.
[7] S. Simmons, The sequence spaces l(pv) and m(p^),
Proc. London Math. Soc. (3) 15 (19^57. 422-36.
[8] E. P. Steiner, Lattices of Topologies on Linear Spaces,
Ph.D. Thesis, 19^3, University of Missouri,
Columbia.
[9] A. E. Taylor, General Theory of Functions and Integration, Blaisdell ^19o:?)/Wew York.
[10] A. E. Taylor, Introduction to Functional Analysis,
John Wiley (195»j, Lonaon.
[11] A. Wilansky, Functional Analysis, Blaisdell (1964),
New York.