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 10: Convergence, continuity and uniform continuity

Let $(X,d)$ and $(C,\rho )$ be metric spaces.

Let $f:X\to C$ be a function.

The function $f:X\to C$ is continuous if $f$ satisfies: if $x\in X$ and $\epsilon \in {ℝ}_{>0}$ then there exists $\delta \in {ℝ}_{>0}$ such that
if $y\in X$ and $d(x,y)<\delta$ then $d(f\left(x\right),f\left(y\right))<\epsilon \text{.}$

The function $f:X\to C$ is uniformly continuous if $f$ satisfies: if $\epsilon \in {ℝ}_{>0}$ then there exists $\delta \in {ℝ}_{>0}$ such that
if $x\in X$ and $y\in X$ and $d(x,y)<\delta$ then $d(f\left(x\right),f\left(y\right))<\epsilon \text{.}$

HW: Second definition The function $f:X\to C$ is continuous if and only if $f$ satisfies if $x\in X$ then $\underset{y\to x}{\text{lim}}f\left(y\right)=f\left(x\right)\text{.}$

HW: Third definition The function $f:X\to C$ is continuous if and only if $f$ satisfies if $x\in X$ and $$\begin{array}{cccc}\stackrel{\rightharpoonup }{x}:& {\mathbb{Z}}_{>0}& ⟶& X\\ & n& ⟼& x_n\end{array}$$ and $\underset{n\to \infty }{\text{lim}}{x}_n=x$
then $\underset{n\to \infty }{\text{lim}}f\left(x_n\right)=f\left(x\right)\text{.}$

HW: Fourth definition The function $f:X\to C$ is continuous if and only if $f$ is continuous as a function between topological spaces i.e., if $f$ satisfies if $V$ is open in $C$ then $f^{-1}\left(V\right)$ is open in $X\text{.}$

Examples

HW Let $f:X\to Y$ and $g:Y\to Z$ be continuous. Show that $g\circ f$ is continuous.

HW Let $A\subseteq X$ and let $f:X\to Y$ be continuous. Show that $$\begin{array}{cccc}g:& A& ⟶& Y\\ & a& ⟼& f\left(a\right)\end{array}$$ is continuous.

HW: Let $f_1:X_1\to Y_1$ and $f_2:X_2\to Y_2$ be continuous. Show that $$\begin{array}{cccc}f:& X_1\times X_2& ⟶& Y_1\times Y_2\\ & (x_1,x_2)& ⟼& (f_1\left(x_1\right),f_2\left(x_2\right))\end{array}$$ is continuous.

HW: Show that $$\begin{array}{ccc}ℝ\times ℝ& ⟶& ℝ\\ (x,y)& ⟼& x+y\end{array}$$ is continuous.

HW: Show that $$\begin{array}{ccc}ℝ& ⟶& ℝ\\ x& ⟼& -x\end{array}$$ is continuous.

HW: Show that $$\begin{array}{ccc}ℝ\times ℝ& ⟶& ℝ\\ (x,y)& ⟼& xy\end{array}$$ is continuous.

HW: Show that $$\begin{array}{ccc}ℝ& ⟶& ℝ\\ x& ⟼& {\displaystyle \frac{x}{1+x^2}}\end{array}$$ is uniformly continuous.

HW: Show that if $f:X\to Y$ is uniformly continuous then $f:X\to Y$ is continuous.

HW: Show that $$\begin{array}{ccc}ℝ& ⟶& ℝ\\ x& ⟼& x^2\end{array}$$ is not uniformly continuous.

Sequences of functions

$f_1,f_2,\dots$ defined by $$\begin{array}{cccc}{f}_n:& [0,1)& ⟶& [0,1)\\ & x& ⟼& x^n\end{array}\text{.}$$

$f_1,f_2,\dots$ defined by $$\begin{array}{cccc}{f}_n:& [0,1]& ⟶& [0,1]\\ & x& ⟼& x^n\end{array}\text{.}$$

$f_1,f_2,\dots$ defined by $$\begin{array}{cccc}{f}_n:& {ℝ}_{\ge 0}& ⟶& {ℝ}_{\ge 0}\\ & x& ⟼& x^n\end{array}\text{.}$$

Let $(X,d)$ and $(C,\rho )$ be metric spaces.

Let $$F=\{\text{functions}\hspace{0.17em}f:X\to C\}$$ and define $d:F\times F\to {ℝ}_{\ge 0}\cup \left\{\infty \right\}$ by $$d(f,g)=\text{sup}\left\{\rho (f\left(x\right),g\left(x\right))\hspace{0.17em}\right|\hspace{0.17em}x\in X\}$$ (warning $d:F\times F\to {ℝ}_{\ge 0}\cup \left\{\infty \right\}$ is not quite a metric).

Let $$\begin{array}{cccc}\stackrel{\rightharpoonup }{f}:& {\mathbb{Z}}_{>0}& ⟶& F\\ & n& ⟼& f_n\end{array}$$ be a sequence of functions from $X$ to $C$ and let $f:X\to C$ be a function.

The sequence $$\begin{array}{cccc}\stackrel{\rightharpoonup }{f}:& {\mathbb{Z}}_{>0}& ⟶& F\\ & n& ⟼& f_n\end{array}$$ converges pointwise to $f$ is $\stackrel{\rightharpoonup }{f}$ satisfies if $x\in X$ and $\epsilon \in {ℝ}_{>0}$ then there exists $N\in {\mathbb{Z}}_{>0}$ such that
if $n\in {\mathbb{Z}}_{>0}$ and $n>N$ then $d(f_n\left(x\right),f\left(x\right))<\epsilon \text{.}$

The sequence $$\begin{array}{cccc}\stackrel{\rightharpoonup }{f}:& {\mathbb{Z}}_{>0}& ⟶& F\\ & n& ⟼& f_n\end{array}$$ converges uniformly to $f$ is $\stackrel{\rightharpoonup }{f}$ satisfies if $\epsilon \in {ℝ}_{>0}$ then there exists $N\in {\mathbb{Z}}_{>0}$ such that
if $x\in X$ and $n\in {\mathbb{Z}}_{>0}$ and $n>N$ then $d(f_n\left(x\right),f\left(x\right))<\epsilon \text{.}$

HW Second definition

The sequence $$\begin{array}{cccc}\stackrel{\rightharpoonup }{f}:& {\mathbb{Z}}_{>0}& ⟶& F\\ & n& ⟼& f_n\end{array}$$ converges pointwise to $f$ is $\stackrel{\rightharpoonup }{f}$ satisfies if $x\in X$ then $\underset{n\to \infty }{\text{lim}}d(f_n\left(x\right),f\left(x\right))=0\text{.}$

The sequence $$\begin{array}{cccc}\stackrel{\rightharpoonup }{f}:& {\mathbb{Z}}_{>0}& ⟶& F\\ & n& ⟼& f_n\end{array}$$ converges uniformly to $f$ if $\underset{n\to \infty }{\text{lim}}d(f_n,f)=0\text{.}$

Notes and References

These are a typed copy of Lecture 10 from a series of handwritten lecture notes for the class Metric and Hilbert Spaces given on August 13, 2014.

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