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 1 and 2: Housekeeping and Proofmachine

Information

(1)	Google: Arun Ram
(2)	Contact and availability
(3)	SSLC Representatives
(4)	Scribe for HWs, Vocabulary and Examples.
(5)	Books: Rubinstein notes and online Notes.
(6)	Schedule – Times away.
(7)	Homework and Exams.
(8)	Proof Machine.

BIG IDEA of the course: CONVERGENCE

A sequence $(x_1,x_2,x_3,\dots )$ converges to $x$ if $(x_1,x_2,\dots )$ satisfies if $\epsilon \in {ℝ}_{>0}$ then there exists $N\in {\mathbb{Z}}_{>0}$ such that if $n\in {\mathbb{Z}}_{>0}$ and $n>N$ then $d(x_n,x)<\epsilon \text{.}$ Write $$\underset{n\to \infty }{\text{lim}}{x}_n=x\text{if}(x_1,x_2,\dots )\hspace{0.17em}\text{converges to}\hspace{0.17em}x\text{.}$$

Rubinstein writes: "Definition 2.8." The sequence $\left\{x_n\right\}$ is said to converge to a point $x$ in $X,$ if for every $\epsilon >0$ there exists a positive integer $k$ such that $d(x_n,x)<x$ for all $n\ge k\text{.}$ In this case we write $$\underset{n\to \infty }{\text{lim}}{x}_n=x\text{or}{x}_n⟶x\text{.}$$ The point $x$ is called the limit of $x_n\text{.}$

$$\begin{array}{rcl}{\displaystyle {ℝ}^n}& =& {\displaystyle \{x=(x_1,x_2,\dots ,x_n)\hspace{0.17em}|\hspace{0.17em}{x}_1,x_2,\dots ,x_n\in ℝ\}}\\ & =& {\displaystyle \{x:{[1,n]}_{\mathbb{Z}}\to ℝ\}}\\ & =& {\displaystyle \left\{\text{functions from}\hspace{0.17em}\{1,2,\dots ,n\}\hspace{0.17em}\text{to}\hspace{0.17em}ℝ\right\}\text{.}}\end{array}$$ Possible norms on ${ℝ}^n\text{:}$ $$\begin{array}{rcl}{\displaystyle \Vert x\Vert }& =& {\displaystyle \sqrt{x_1^2+x_2^2+\cdots +x_n^2},}\\ {\displaystyle {\Vert x\Vert }_p}& =& {\displaystyle {({\mid x_1\mid }^p+{\mid x_2\mid }^p+\cdots +{\mid x_p\mid }^p)}^{\frac{1}{p}},}\\ {\displaystyle {\Vert x\Vert }_{\infty }}& =& {\displaystyle \text{sup}\{\mid x_1\mid ,\mid x_2\mid ,\dots ,\mid x_n\mid \}\text{.}}\end{array}$$

$$\begin{array}{rcl}{\displaystyle {ℝ}^{\infty }}& =& {\displaystyle \{x=(x_1,x_2,\dots )\hspace{0.17em}|\hspace{0.17em}{x}_i\in ℝ\}}\\ & =& {\displaystyle \{\text{sequences}\hspace{0.17em}{x}_1,x_2,\dots \hspace{0.17em}\text{in}\hspace{0.17em}ℝ\}}\\ & =& {\displaystyle \{x:{\mathbb{Z}}_{>0}\to ℝ\}}\\ & =& {\displaystyle \left\{\text{functions from}\hspace{0.17em}\{1,2,\dots \}\hspace{0.17em}\text{to}\hspace{0.17em}ℝ\right\}\text{.}}\end{array}$$ Possible norms on ${ℝ}^{\infty }\text{:}$ $$\begin{array}{rcl}{\displaystyle \Vert x\Vert }& =& {\displaystyle {\left(\sum _{i=1}^{\infty }{\mid x_i\mid }^2\right)}^{\frac{1}{2}}\text{gives}{\ell }^2,}\\ {\displaystyle {\Vert x\Vert }_p}& =& {\displaystyle {\left(\sum _{i=1}^{\infty }{\mid x_i\mid }^p\right)}^{\frac{1}{p}}\text{gives}{\ell }^p,}\\ {\displaystyle {\Vert x\Vert }_{\infty }}& =& {\displaystyle \text{sup}\{\mid x_1\mid ,\mid x_2\mid ,\dots \}\text{gives}{\ell }^{\infty }\text{.}}\end{array}$$

Consider $\{f:[0,1]\to ℝ\}$ or $\{f:X\to ℝ\}\text{.}$ Can we put norms on these to get $L^2\left(X\right),$ $L^p\left(X\right),$ $L^{\infty }\left(X\right)\text{?}$

Notes and References

These are a typed copy of Lecture 1 and 2 from a series of handwritten lecture notes for the class Metric and Hilbert Spaces given on July 29 and July 30, 2014.

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