Download 1 Metric spaces

1
Metric spaces
The sets that most often occur in the study of pdes consist of subsets of Rn ; n 2
N and sets of various kinds of functions. In order to do analysis (i.e. carry
out limiting processes) with elements of these sets it is necessary to introduce a
concept of "distance" between elements so that we can determine how "close"
two elements are to each other. A "metric" de…ned on a set enables this to be
done.
1.1 De…nition: A metric on a set X is a function d : X
that
(i) non-negativity
d (x; y)
X ! R such
0 and d (x; y) = 0 , x = y
(ii) symmetry
d (y; x) = d (x; y) ;
(iii) the triangle inequality
d (x; z)
d (x; y) + d (y; z) .
The number d (x; y) is called the distance from x to y. A pair (X; d) is called
a metric space.
Note: If there is no ambiguity write ‘the metric space X’ and also use the
same symbol d for the metric on di¤erent spaces.
Examples: (i) The real line R and complex plane C are metric spaces
under the standard metric d (x; y) = jx yj.
(ii) Rn and Cn (i.e. R2n ) are metric spaces under the Euclidean metric
0
d (x; y) = @
n
X
j=1
jxj
where x = (x1 ; :::; xn ), y = (y1 ; :::; yn ).
1.1
11=2
2
yj j A
Subsets of a metric space
We can use the metric to de…ne various kinds of subsets of X which we will
need in the analysis of pdes.
1.2 De…nition: (i) A closed ball (of radius r, about a) is a set of the form
Bc (a; r) = fx 2 X : d (a; x)
rg
where a 2 X, r > 0.
(ii) An open ball (of radius r, about ound a) is a set of the form
Bo (a; r) = fx 2 X : d (a; x) < rg
1
(iii) A sphere is the set fx 2 X : d (a; x) = rg.
1.3 De…nition: (i) A neighbourhood N , of a point a 2 X is a subset of
X such that N contains a ball around a.
a is called an interior point of N .
(ii) The interior int (A) of A is the set of all interior points of A.
(iii) An open set is such that every point of it is an interior point.
(iv) A closure point of a subset A is a point a such that every neighbourhood
of a has a non-empty intersection with A.
(v) The closure A of A is the set of all closure points of A.
(vi) A closed set contains all of its closure points.
(vii) The boundary of A is the set A n int (A) consisting of those points each
of whose neighbourhoods has a non-empty intersection with both A and X n A.
(viii) A subset A is dense in X if A = X.
Note: A neighbourhood is often required to be an open set.
1.4 De…nition: A sequence fxn g in X converges to a point a 2 X if
8" > 0, 9n0 such that d (xn ; a) < " 8n n0 .
Write xn ! a or d (xn ; a) ! 0 as n ! 1.
Equivalently, fxn g converges to a if 8 neighbourhoods N of a 9n0 such
that xn 2 N 8 n n0 (and say that xn is eventually in N ).
A sequence converges to at most one point a, called the limit of fxn g, a
= limn xn .
1.5 De…nition (i) : The map f : X ! Y between metric spaces X; Y is
continuous at a 2 X if 8 neighbourhoods N of f (a) 9 a neighbourhood M of
a such that f (M ) N .
(ii) f is said to be continuous on X or continuous if f is continuous at
each point of X.
(i) f is a homeomorphism if it is one-to-one, onto, and both f and f 1
are continuous. X and Y are said to be homeomorphic under f .
1.6 Proposition: (i) open balls are open;
(ii) closed balls are closed;
(iii) int (A) is the largest open set inside A;
(iv) A is the smallest closed set containing A;
(v) the neighbourhood, sequential, "
and open/closed de…nitions of continuity are equivalent;
(vi) a homeomorphism f : X ! Y sets up a one-to-one correspondence between
open sets, closed sets and convergent sequences in X and Y .
Two metrics d; e on a set X are equivalent if they give the same open sets,
i.e. the identity map x 7 ! x of (X; d) to (X; e) is a homeomorphism. This is
so , each ball in the d-metric (‘d-ball’) contains a concentric e-ball and vice
versa.
1.2
Subspaces and product spaces
A subset A of a metric space (X; d) becomes a metric space ( a subspace of
(X; d)) under the relative metric dA obtained by restricting d to A A.
2
c
o
Balls in (A; dA ) are given by BA
(a; r) = A \ B c (a; r) ; BA
(a; r) = A \
B (a; r), where a 2 A. Thus, the open (closed) sets in (A; dA ) are the sets
A \ S where S is open (closed) in (X; d). If f : (X; d) ! (Y; e) is continuous
then the restriction f jA : (A; dA ) ! (Y; e) is continuous.
1.6 De…nition: Let (X; d), (Y; e) be metric spaces. Two standard metrics
for the product X Y are
o
((x; y) ; (x0 ; y 0 )) = d (x; x0 ) + e (y; y 0 ) ;
((x; y) ; (x0 ; y 0 )) = maxfd (x; x0 ) ; e (y; y 0 )g:
The metrics , are equivalent: given a -ball of radius r0 , the concentric -ball
of radius r0
-ball; given a -ball of radius r0 , the concentric -ball of radius
r0 =2
-ball.
1.7 Proposition: A sequence f(xn ; yn )g converges to (a; b) in (X Y; ) or
(X Y; ) , xn ! a in (X; d) and yn ! b in (Y; e).
The corresponding result holds for the analogous metrics on an arbitrary
…nite product X1 ::: Xn of metric spaces. Thus, convergence in the product
space is coordinate-wise convergence.
1.3
Uniform continuity
1.8 De…nition: Let (X; d), (Y; e) be metric spaces. The map f : X ! Y is
uniformly continuous if 8" > 09 > 0 such that e (f (x) ; f (y)) < " 8 x; y 2 X
such that d (x; y) < .
Example: The map x 7! x2 on R with the usual metric is continuous but
not uniformly continuous.
1.3.1
Metric Properties of subsets
1.9 De…nition: Let A; B be nonempty subsets of (X; d). The distance from
A to B is de…ned by d (A; B) = inf fd (a; b) : a 2 A; b 2 Bg. When A consists of
a single point fag write d (a; B) = inf fd (a; b) : b 2 Bg.
1.10 Lemma: The real-valued function x 7 ! d (x; B) is uniformly continuous on X.
Taking B to consist of a single point shows that the map x 7 ! d (x; b) is
uniformly continuous.
1.11 De…nition: A subset A is bounded if its diameter diam (A) =
sup fd (x; y) : x; y 2 Ag is …nite.
Note: A is bounded, A lies inside some ball.
1.3.2
Completeness
1.12 De…nition: A sequence fxn g in a metric space (X; d) is said to be a
Cauchy sequence if 8" > 09n0 such that d (xm ; xn ) < " 8m; n n0 .
1.13 Lemma: If a Cauchy sequence fxn g has a convergent subsequence
fxnk g convergent to a point x, then fxn g also converges to x.
Every convergent sequence is Cauchy.
3
1.14 De…nition: (X; d) is complete if every Cauchy sequence is convergent.
1.15 Theorem: R and C are complete.
Note: fxn g is Cauchy , diam fxk : k ng ! 0 as n ! 1.
1.16 Theorem: Cantor Let fAn g be a decreasing sequence of non-empty
closed sets in \
a complete metric space (X; d) such that diam fAn g ! 0 as
n ! 1. Then
An contains exactly one point.
n
1.17 Proposition: Let A be a subset of a metric space (X; d) such that the
subspace (A; dA ) is complete. Then A is closed. Conversely, let A be a closed
subset of a complete metric space (X; d) then (A; dA ) is complete.
(1.17) may be a convenient way of showing a given metric space is not complete: embed it as a subspace of a larger metric space (with the metric of the
subspace a restriction of the larger space) and show that it is not closed in this
space.
1.18 Proposition: The product of complete metric spaces (X; d), (Y; e) is
complete under either standard metric in (1.6).
1.3.3
Compactness
1.19 De…nition: A subset C of a metric space (X; d) is called compact if every
family of open sets which covers C (i.e. whose union contains C) has a …nite
family which also covers C.
Notes (i): This de…nition generalises to topological spaces, see below.
(ii) Open sets in C are precisely the sets U \ C where U is open in X.
Thus, C is compact in (X; d) , C is compact in (C; dC ), i.e. compactness is a
property of C and not of its relationship with X.
The following analogue of Cantor’s theorem follows from 1.19.
1.20 Proposition: Let fAn g be a decreasing sequence of nonempty closed
sets in a compact metric space then \n An is nonempty.
The next theorem derives two conditions equivalent to compactness in a
metric space. Condition (ii) is often taken as an alternative de…nition.
1.21 De…nition:Given a set A in a metric space (X; d) then
(i) An "-net for A is a …nite set of points fx1 ; :::xn g such that each of the
following equivalent statements hold,
(a) each point of A is distant at most " from some xj , j = 1; :::; n,
(b) the closed balls of radius " about the xj cover A.
(ii) A is totally bounded or precompact if one of the following equivalent statements hold
(a) it possesses an "-net for each " > 0,
(b) A can be covered by …nitely many sets A1 ; :::; An of diameter
":
Notes (1): 1.21 (ii) (a), (b) are equivalent since "-balls have diameter at
most ":
4
(2) In 1.21 (ii) (b) the Ak may be taken closed (replacing Ak by Ak ) or
disjoint [replacing Ak by Ak n (A1 [ ::: [ Ak 1 )], but not generally both together.
(3) An "-net for A is also an "-net for A ; if A is totally bounded so is A .
1.22 Theorem: The following are equivalent for a metric space (X; d),
(i) X is compact;
(ii) Every sequence in X has a convergent subsequence (the Bolzano- Weierstrass property)
(iii) X is complete and totally bounded.
1.23 Corollary: A subset of a complete metric space has compact closure
, it is totally bounded.
Remark: Every bounded subset of R (or Rn ) has an "-net for each " > 0.
This implies the
1.24 Heine-Borel Theorem: A subset of R (or Rn ) is compact , it is
closed and bounded.
In any metric space, a compact subset is closed (because complete) and
bounded (totally bounded sets are bounded).
1.25 Theorem: Let (X; d) and (Y; e) be compact metric spaces then X Y
is compact (under either standard metric).
One …nal de…nition (useful when using power series and multiple and line
integrals):
1.26 De…nition: (i) A set S in a metric space is said to be connected if
it is not the union of non-empty disjoint sets.
(ii) A set S is called disconnected if it is not connected.
2
Topological spaces
We give a very brief introduction to toplogical spaces which are often used in
advanced books on partial di¤erential equations. Such spaces represent an an
alternative way of developing a theory of limits and continuity which is more
general than that of a metric space. Essentially, the concept of ‘metric’ is
replaced by that of ‘open set’. Now, in a metric space, the class of open sets
satisfy the following properties,
(1) X and ; are open (the latter by convention to allow property (3) below).
(2) The union of any family of open sets is open.
(3) The intersection of two (and, by induction, …nitely many) open sets is
open.
(4) Given distinct points x; y 2 X there exist disjoint open sets U; V with
x 2 U; y 2 V:
1.27 De…nition: (i) A topology for a non-empty set X is a family T of
subsets of X, called open (or T-open) sets such that properties (1), (2), (3)
above hold.
(ii) A pair (X; T) is called a topological space.
(iii) If (4) also holds the space and topology are called Hausdor¤.
Notes (i): T is often omitted and X called a topological space.
5
(ii) Every metric space X is a Hausdor¤ topological space when X is given the
metric topology, i.e.the class of sets open in the sense of defn. 1.3(iii).
In a topological space (X; T) a set N is said to be a neighbourhood of a
point a (and a is said to be interior to N ) if 9 an open set U 2 T, such that
a 2 U N.
1.28 Theorem: Sets that are open in the sense of 1.3(iii) are exactly the
open sets of 1.27.
De…ne exactly as for a metric space the concepts of closure point, closure,
interior, closed, boundary, dense, continuous at a point for maps from
one topological space to another, continuous, homeomorphism and the elementary properties are unaltered.
1.29 Theorem: f : X ! Y is continuous , U is open in Y ) f 1 (U ) is open
in X , F is closed in Y ) f 1 (F ) is closed in X.
Convergence of a sequence is de…ned as in 1.4. In general a sequence
may converge to more than one point; the Hausdor¤ property (4) ensures that
limits are unique. Let U; V be disjoint open neighbourhoods of points x; y and
let xn ! x. Then for su¢ ciently large n, xn 2 U and so 2
= V , so xn 9 y.
Reference:
[1] J.D. Pryce, Basic Methods of Linear Functional Analysis, Hutchinson,
London, 1973.
6