Metric and Hilbert Spaces

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 4 November 2014

Lecture 45: Product spaces and equivalent metrics

Examples of spaces

Subspaces and product spaces, $B(V,W),$ Function spaces.

Subspaces

HW:

(a)	Let $X$ be a normed vector space. Let $V\subseteq X$ be a subspace. Show that $V$ is a normed vector space with the same norm [Bressan, Ch 2 P4].
(b)	Let $(X,d)$ be a metric space. Let $Y\subseteq X$ be a subset. Show that $(Y,d)$ is a metric space [Rubinstein Example 2 to Def. 2.1].
(c)	State and prove similar statements for topological spaces and for inner product spaces (see Rubinstein Theorem 2.21?)

Product spaces

HW: Let $X$ and $Y$ be Banach spaces.

(a)	Prove that $X\times Y$ with $$\Vert (x,y)\Vert =\text{max}\{\Vert x\Vert ,\Vert y\Vert \}$$ is a Banach space [Bressan Ch 2 P2].
(b)	Prove that $X\times Y$ with $$\Vert (x,y)\Vert ={({\Vert x\Vert }^2+{\Vert y\Vert }^2)}^{\frac{1}{2}}$$ is a normed vector space. Is it a Banach space?
(c)	Are $X\times Y$ from (a) and $X\times Y$ from (b) the same as topological spaces? [Bressan Ch 5 Ex 2].

HW: Let $(X_1,d_1)$ and $(X_2,d_2)$ be metric spaces. Let $Y=X_1\times X_2$ and define $$\begin{array}{rcl}{\displaystyle d((x_1,x_2),(y_1,y_2))}& =& {\displaystyle d_1(x_1,y_1)+d_2(x_2,y_2),}\\ {\displaystyle \rho ((x_1,x_2),(y_1,y_2))}& =& {\displaystyle \text{max}\{d_1(x_1,y_1),d_2(x_2,y_2)\},}\\ {\displaystyle \sigma ((x_1,x_2),(y_1,y_2))}& =& {\displaystyle \sqrt{d_1{(x_1,x_2)}^2+d_2{(x_2,y_2)}^2}\text{.}}\end{array}$$

(a)	Show that $(Y,d),(Y,\rho )$ and $(Y,\sigma )$ are metric spaces.
(b)	Show that $(Y,d),(Y,\rho )$ and $(Y,\sigma )$ are the same as topological spaces.

[Rubinstein Example (3) to Def. 2.1 and Example 2 to Def. 2.11].

$(0,1)$ is homeomorphic to $ℝ\text{.}$ $$\begin{array}{c} \end{array}$$ $$\begin{array}{ccc}(0,1)\hspace{0.17em}\text{is bounded}& & (0,1)\hspace{0.17em}\text{is not complete}\\ ℝ\hspace{0.17em}\text{is not bounded}& & ℝ\hspace{0.17em}\text{is complete}\end{array}$$ So boundedness and completeness are not topological properties.

Let $X$ be a set and let $d_1:X\times X\to {ℝ}_{\ge 0}$ and $d_2:X\times X\to {ℝ}_{\ge 0}$ be metrics on $X\text{.}$

The metrics $d_1$ and $d_2$ are Lipschitz equivalent if there exist $c_1,c_2\in {ℝ}_{>0}$ such that if $x,y\in X$ then $c_1{d}_1(x,y)\le d_2(X,Y)\le c_2{d}_1(x,y)\text{.}$

Let $X$ be a set and let $d_1$ and $d_2$ be Lipschitz equivalent metrics on $X\text{.}$ Show that $d_1$ and $d_2$ produce the same topology on $X\text{.}$

		Sketch of proof.
	To show: If $x\in X$ and $\epsilon \in {ℝ}_{>0}$ then $$B_{\epsilon }^2\left(x\right)=\{y\in X\hspace{0.17em}|\hspace{0.17em}{d}_2(x,y)<\epsilon \}$$ is $d_1\text{-open.}$ Assume $x\in X$ and $\epsilon \in {ℝ}_{>0}\text{.}$ To show: There exists $\delta \in {ℝ}_{>0}$ such that $B_{\delta }^{\prime }\left(x\right)\subseteq B_{\epsilon }^2\left(x\right)$ where $$B_{\delta }^1\left(x\right)=\{y\in X\hspace{0.17em}|\hspace{0.17em}{d}_1(x,y)<\delta \}\text{.}$$ Let $\delta =C_1\epsilon \text{.}$ $\square$

Notes and References

These are a typed copy of Lecture 45 from a series of handwritten lecture notes for the class Metric and Hilbert Spaces given on October 18, 2014.

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