Download Locally Convex Vector Spaces I: Basic Local Theory

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Gabriel
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LCVS I
Locally Convex Vector Spaces I:
Basic Local Theory
Notes from the Functional Analysis Course (Fall 07 - Spring 08)
Convention. Throughout this note K will be one of the fields R or C, and all vector
spaces are over K.
Definition. A locally convex vector space is a pair (X , T) consisting of a vector space X
and linear topology T on X , which is locally convex, in the sense that:
(lc) every x ∈ X possesses a fundamental system of convex neighborhoods.
A locally convex topological vector space is a locally convex vector space, whose topology
is Hausdorff. Since convexity is translation invariant, for a linear topology T, the local
convexity condition (lc) needs only to be verified at x = 0.
The following result is a locally convex analogue of Proposition 2.B from TVS I.
Proposition 1. In a locally convex space X there exists a basic neighborhood system for
0, which consists of balanced open convex sets.
Proof. What we need to prove is that:
(∗) for every neighborhood N of 0, there exists a balanced open convex set A ⊂ N .
First of all, by definition, there exists a convex neighborhood V of 0, such that V ⊂ N .
Secondly, by Proposition 2.B from TVS I, there exists some open balanced set B ⊂ V. Now
we are done, by taking A = conv(B). (By CW I, A is balances open convex. Since V is
convex and contains B, it will also contain A.)
The next result is a locally convex analogue of the Corollary to Theorem 1 from TVS
I. It should ne noted, however, that the hypotheses are somehow “more economical” (the
collection C could be, for instance, finite), and the conclusion is slightly weaker.
Theorem 1. Suppose X is a vector space and C is a collection of balanced1 absorbing
convex sets. Then there exists a unique locally convex linear topology T on X , such that the
collection
U = {εC : C ∈ C, ε > 0}
constitutes a fundamental system of T-neighborhoods of 0.
Proof. Define the collection V consisting of finite intersections of sets in U. We wish to
apply Corollary for Theorem 1 from TVS I, so we need to check the following:
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The fact that C is balanced forces C 3 0.
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(i) V is a filter, and all sets in V contain 0;
(ii) every V ∈ V is absorbing;
(iii) for every V ∈ V, there exists W ∈ V, such that W + W ⊂ V;
(iv) every V ∈ V is balanced.
Condition (i) is trivial, since all sets in U contain 0.
Condition (iv) is also clear, since every set in U is balanced, and (arbitrary) intersections
of balanced sets are balanced.
Condition (ii) follows from Exercise 2 in CW II, since all sets in U are convex and
absorbing.
To check condition (iii), start with some V ∈ V, so there exists C1 , . . . , Cn ∈ C and
ε1 , . . . , εn > 0, such that
V = (ε1 C1 ) ∩ · · · ∩ (εn Cn ).
Consider then the set
W = ( ε21 C1 ) ∩ · · · ∩ ( ε2n Cn ),
which again belongs to V. Obviously 2W ⊂ V, and since W is convex, we also have
2W = W + W, so we are done.
Having checked the above conditions, we invoke the above mentioned result, to conclude
that there exists a unique linear topology T that has V as a basic system of neighborhoods
for 0. By construction, T is also the unique linear topology which has U as a fundamental
system of neighborhoods for 0. Finally, since all sets in V are convex, it follows that T is
indeed locally convex.
Remark 1. With C and T as in Theorem 1, the following are equivalent:
(i) T is Hausdorff;
\
(ii)
(εC) = {0}.
C∈C
ε>0
This is quite clear
T since, using the notations from the proof, condition (ii) is equivalent to
the condition V∈V V = {0}, which in turn is (see Theorem 1 in TVS I) equivalent to (i).
Exercise 1. Suppose that X is locally convex. Prove that the collection of all closed,
convex, balanced neighborhoods of 0, constitutes a basis system of neighborhoods of 0. That
is, for every neighborhood V of 0, there exists a closed, convex, balanced neighborhood W
of 0, such that W ⊂ V.
Exercises 2-4. Suppose X is an infinite dimensional vector space.
2. Let C be the collection of all absorbing balanced convex subsets of X , and let T be the
corresponding locally convex linear topology described by Theorem 1. Prove that:
(i) T is Hausdorff;
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(ii) if Y is a locally convex vector space, then all linear maps T : X → Y are Tcontinuous;
(iii) all linear subspaces are T -closed;
(iv) T is the strongest locally convex linear topology on X .
3. Consider the algebraic dual of X , i.e. the space
X 0 = {φ : X → K : φ linear },
and define, for φ ∈ C, the set Cφ = {x ∈ X : |φ(x)| < 1}. Let C0 = {Cφ : φ ∈ X 0 },
and let T0 be the corresponding locally convex linear topology described by Theorem
1. Prove that:
(i) T0 is Hausdorff;
(ii) all maps φ ∈ X 0 are T’-continuous;
(iii) all linear subspaces are T ’-closed.
4. Show that the topology T0 is strictly weaker than T, i.e. there exist T-open sets which
are not T0 -open. (Hint: Every T0 - neighborhood contains an infinite dimensional linear
subspace. There are, however, T-neighborhoods of 0 without this property.)
Exercises 5-6 Let 0 < p < 1. We denote by Dp the metric on `p , defined (see TVS I) by
Dp (x, y) =
∞
X
|xn − yn |p , ∀ x = (xn ), y = (yn ) ∈ `p .
n=1
For every ρ > 0 we define the ball
Bρ = {x ∈ `p : Dp (x, 0) < ρ}.
5. Show that: sup Dp (x, 0) : x ∈ conv(Bρ ) = ∞, ∀ ρ > 0.
6. Use the above Exercise to prove that the metric topology on `p is not locally convex.
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