Complete Metric spaces and Function spaces

7 Complete Metric spaces and Function spaces

7.1 Complete Metric Spaces

7.1.1 Cauchy sequence (x_{n}) in (X,d) ( a sequence (x_{n}) of points of X, a metric space ( X,d)) : it has the property that given \epsilon > 0 , there is an integer N s.t. d(x_{n}, x_{m}) < \epsilon whenever n, m \ge N

7.1.2 Complete ( a metric space (X,d) ) : Every Cauchy sequence in X converges.

7.1.2.1 If X is complete under the metric d, then X is complete under the standard bounded metric corresponding to d.

7.1.3 (Lem 43.1) : A metric space X is complete if every Cauchy sequence in X has a convergent subsequence.

7.1.4 (Thm 43.2) Euclidean space \mathbb{R}^{k} is complete in either of its usual metrics, the Euclidean metric d or the square metric \rho.

7.1.4.1 Pf) (Thm 28.2)

7.1.5 (Lem 43.3) Let X be the product space X = \prod X_{\alpha} ; let x_{n} be a sequence of points of X. Then x_{n} -> X iff \pi_{\alpha} (\mathbf{x}{n}) -> \pi{\alpha} (\mathbf{x}) for each \alpha.

7.1.6 (Thm 43.4) There is a metric for the product space \mathbb{R}^{\omega} relative to which \mathbb{R}^{\omega} is complete.

7.1.6.1 Pf) D(x,y) = sup{\bar{d}(x_{i}, y_{i}) / i}

7.1.7 Uniform metric on Y^{J} corresponding to the metric d on Y : \bar{\rho} (\mathbf{x}, \mathbf{y}) = sup{\bar{d} (x_{\alpha}, y_{\alpha}) | \alpha \in J)

7.1.7.1 Let (Y,d) be a metric space; let \bar{d} (a,b) = min{d(a,b), 1} be the standard bounded metric on Y derived from d. If \mathbf{x} = (x_{\alpha}){a \in J} and \mathbf{y} = (y{\alpha})_{a \in J} are points of the cartesian product Y^{J}.

7.1.8 (Thm 43.5) If the space Y is complete in the metric d, then the space Y^{J} is complete in the uniform metric \bar{\rho} corresponding to d.

7.1.8.1 Pf) (if (Y,d) is complete, so is (Y, \bar{d}))

7.1.9 Bounded ( a function f : X -> Y) : its image f(X) is a bounded subset of the metric space (Y,d)

7.1.10 (Thm 43.6) Let X be a topological space and let (Y,d) be a metric space. The set \mathcal{C} (X,Y) of continuous functions is closed in Y^{X} under the uniform metric. So is the set \mathcal{B} (X,Y) of bounded functions. Therefore, if Y is complete, these spaces are complete in the uniform metric.

7.1.10.1 Pf) (Uniform limit theorem, Thm 21.6)

7.1.11 Sup metric ( a metric \rho) : \rho ( f, g) = sup { d(f(x), g(x)) | x \in X} If (Y,d) is a metric space, one can define another metric on the set \mathcal{B} (X,Y) of bounded functions from X to Y . \rho is well-defined, for the set f(X) U g(X) is bounded if both f(x) and g(X) are.

7.1.11.1 If X is a compact space, then every continuous function f: X -> Y is bounded, hence the sup metric is defined on \mathcal{C} (X, Y)

7.1.12 (Thm 43.7) Let (X,d) be a metric space, There is an isometric imbedding of X into a complete metric space.

7.1.12.1 Isometric imbedding of X in Y : an imbedding f:X->Y have the property that for every pair of points x_{1}, x_{2} of X, d_{Y} (f(x_{1}), f(x_{2})) = d_{X} (x_{1}, x_{2})

7.1.12.2 Pf) ( a fixed point : an element of the function’s domain that is mapped to itself by the function)

7.1.13 Completion \bar{h(X)} of X ( a metric space X) : a subspace \bar{h(x)} of Y, if h: X->Y is an isometric imbedding of X into a complete metric space Y

7.1.13.1 Completion of X is uniquely determined up to an isometry.

7.2 Space-filling Curve

7.2.1 (Thm 44.1) Let I = [0,1]. There exists a continuous map f: I -> I^{2} whose image fills up the entire square I^{2}.

7.2.1.1 Pf) (closed -> complete -> \mathcal{C}(I,I^{2}) is complete)

7.3 Compactness in Metric spaces

7.3.1 Every compact metric space is complete

7.3.1.1 Pf) (Lem 43.1)

7.3.2 Totally bounded ( a metric spacx (X,d)) : for every \epsilon >0, there is a finite covering of X by \epsilon -balls.

7.3.3 (Thm 45.1) A metric space (X,d) is compact iff it is complete and totally bounded.

7.3.3.1 Pf) (Every compact metric space is complete)

7.3.4 Equicontinuous \mathcal{F} at x_{0} (a subset \mathcal{F} of the function space \mathcal{C} (X,Y), a x_{0} \in X) : Let (Y,d) be a metric space. If given \epsilon >0, there is a neighborhood U of x_{0} s.t. for all x \in U and all f \in \mathcal{F}, d(f(x), f(x_{0})) < \epsilon .

7.3.4.1 Equicontinuous (a set \mathcal{F}) : the set \mathcal{F} is equicontinuous at x_{0} for each x_{0} \in X.

7.3.5 (Lem 45.2) Let X be a space; Let (Y,d) be a metric space. If the subset \mathcal{F} of \mathcal{C} (X,Y) is totally bounded under the uniform metric corresponding to d, then \mathcal{F} is equicontinuous under d.

7.3.6 (Lem 45.3) Let X be a pace; let (Y,d) be a metric space; assume X and Y are compact. If the subset \mathcal{F} of \mathcal{C}(X,Y0 is equicontinuous under d, then \mathcal{F} is totally bounded under the uniform and sup metrics corresponding to d.

7.3.7 Pointwise bounded \mathcal{F} under d ( a subset \mathcal{F} of the function space \mathcal{C} (X,Y) , a metric space (Y,d)) : for each x \in X, the subset \mathcal{F}_{a} = {f(a) | f \in \mathcal{F}} of Y is bounded under d.

7.3.8 (Ascoli’s Theorem, classical version) (Thm 45.4) : Let X be a compact space; Let (\mathbb{R},d) denote euclidean space in either the square metric or the Euclidean metric; give \mathcal{C} (X,\mathbb{R}^{n}) the corresponding uniform topology. A subspace \mathcal{F} of \mathcal{C} (X,\mathbb{R}^{n}) has compact closure iff \mathcal{F} is equicontinuous and pointwise bounded under d.

7.3.8.1 Pf) (Thm 45.1) (Lem 45.2) (Lem 45.3)

7.3.8.2 (Cor) Let X be compact; let d denote either the square metric or the Euclidean metric on \mathbb{R}^{n} ; give \mathcal{C} (X,\mathbb{R}^{n}) the corresponding uniform topology. A subspace \mathcal{F} of \mathcal{C} (X,\mathbb{R}^{n}) is compact iff it is closed, bounded under the sup metric \rho, and equicontinuous under d.

7.4 Pointwise and Compact convergence

7.4.1 Topology of pointwise convergence : given a point x of the set X and an open set U of space Y, let S(x,U) = {f | f \in Y^{X} and f (x) \in U}. The sets S(x,U) are a subbasis for topology on Y^{X}.

7.4.1.1 Also called ‘point-open topology’

7.4.2 (Thm 46.1) A sequence f_{n} of functions converges to the function f in the topology of pointwise convergence iff for each x in X, the sequence f_{n} (x) of points of Y converges to the point f(x).

7.4.2.1 Pf) (Lem 43.3)

7.4.3 Topology of compact convergence : Let (Y,d) be a metric space; let X be a topological space. Given an element f of Y^{X}, a compact subspace C of X, and a number \epsilon >0, let B_{C} (f, \epsilon) denote the set of all those elements g of Y^{X} for which sup{d(f(x), g(x)) | x \in C} < \epsilon. The sets B_{C} (f, \epsilon ) form a basis for a topology on Y^{X}.

7.4.3.1 Also called ‘Topology of uniform convergence on compact sets’

7.4.4 (Thm 46.2) A sequence f_{n} : X -> Y of functions converges to the function f in the topology of compact convergence iff for each compact subspace C of X, the sequence f_{n} | C converges uniformly to f | C.

7.4.5 Compactly generated (a space X) : it satisfies the following condition.

7.4.5.1 A set A is open in X if A \cap C is open in C for each compact subspace C of X.

7.4.6 (Lem 46.3) If X is locally compact, or if X satisfies the first countability axiom, then X is compactly generated.

7.4.7 (Lem 46.4) If X is compactly generated, then a function f: X->Y is continuous if for each compact subspace C of X, the restricted function f | C is continuous.

7.4.8 (Thm 46.5) Let X be a compactly generated space; let (Y,d) be a metric space. Then \mathcal{C} (X,Y) is closed in Y^{X} in the topology of compact convergence.

7.4.8.1 Pf) ( Uniform limit theorem)

7.4.8.2 (Cor 46.6) Let X be a compactly generated space; let (Y,d) be a metric space. If a sequence of continuous functions f_{n} : X -> Y converges to f in the topology of compact convergence, then f is continuous.

7.4.9 (Thm 46.7) Let X be a space; let (Y,d) be a metric space. For the function space Y^{X}, one has the following inclusions of topologies :

7.4.9.1 (Uniform) \supseteq (Compact convergence) \supseteq (Pointwise convergence)

7.4.9.2 If X is compact, the first two coincide, and if X is discrete, the second two coincide.

7.4.10 Compact-open topology : Let X and Y be topological spaces. If C is a compact subspace of X and U is an open subset of Y, define S(C,U) = {f | f \in \mathcal{C} (X,Y) and f(C) \subset U}. The sets S(C,U) forms a subbasis for a topology on \mathcal{C} (X,Y).

7.4.11 (Thm 46.8) Let X be a space and let (Y,d) be a metric space. On the set \mathcal{C} (X,Y), the compact-open topology and the topology of compact convergence coincide.

7.4.11.1 (Cor 46.9) Let Y be a metric space. The compact convergence topology on \mathcal{C} (X,Y) does not depend on the metric of Y. Therefore if X is compact, the uniform topology on \mathcal{C} (X,Y) does not depend on the metric of Y.

7.4.12 (Thm 46.10) Let X be a locally compact Hausdorff; let \mathcal{C} (X,Y) have the compact-open topology. Then the map e: X \times \mathcal{C} (X,Y) -> Y defined by the equation e(x,f) = f(x) is continuous.

7.4.12.1 Evaluation map : the map e

7.4.13 Map F induced by f ( a function f : X \times Z -> Y) : There is a corresponding function F : Z -> \mathcal{C} (x<Y), defined by the equation (F(z))(x) = f(x,z). Conversely, given F : Z -> \mathcal{C} (X,Y), this equation defines a corresponding function f : X \times Z -> Y.

7.4.14 (Thm 46.11) Let X and Y be spaces; give \mathcal{C} (X,Y) the compact-open topology. If f: X \times Z -> Y is continuous, then so is the induced function F: Z -> \mathcal{C} (X,Y). The converse holds if X is locally compact Hausdorff.

7.4.14.1 Pf) (Tube lemma)

7.4.15 Homotopic (two functions f and g in \mathcal{C} (X,Y) ) : There is a continuous map h : X \times [0,1] -> Y s.t. h(x,0) = f(x) and h(x,1) = g(x) for each x \in X.

7.4.15.1 Homotopy between f and g : the map h

7.5 Ascoli’s Theorem

7.5.1 (Ascoli’s Theorem) (Thm 47.1) : Let X be a space and let (Y,d) be a metric space. Give \mathcal{C} (X,Y) the topology of compact convergence; Let \mathcal{F} be a subset of \mathcal{C} (X,Y).

7.5.1.1 If \mathcal{F} is equicontinuous under d and the set \mathcal{F}_{a} = { f(a) | f\in \mathcal{F} } has compact closure for each a \in X, then \mathcal{F} is contained in a compact subspace of \mathcal{C} (X,Y).

7.5.1.2 The converse holds if X is locally compact Hausdorff.

7.5.1.3 Pf) (Tychonoff Theorem) (Thm 46.8) (Thm 46.10) (Thm 45.1) (Lem 45.2)