Topological Spaces

62 minute read

posted: 01-Aug-2025 & updated: 03-Aug-2025

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Introduction

Preamble

Notations

sets of numbers
- $\naturals$ - set of natural numbers
- $\integers$ - set of integers
- $\integers_+$ - set of nonnegative integers
- $\rationals$ - set of rational numbers
- $\reals$ - set of real numbers
- $\preals$ - set of nonnegative real numbers
- $\ppreals$ - set of positive real numbers
- $\complexes$ - set of complex numbers
sequences $\seq{x_i}$ and the like
- finite $\seq{x_i}_{i=1}^n$, infinite $\seq{x_i}_{i=1}^\infty$ - use $\seq{x_i}$ whenever unambiguously understood
- similarly for other operations, e.g., $\sum x_i$, $\prod x_i$, $\cup A_i$, $\cap A_i$, $\bigtimes A_i$
- similarly for integrals, e.g., $\int f$ for $\int_{-\infty}^\infty f$
sets
- $\compl{A}$ - complement of $A$
- $A\sim B$ - $A\cap \compl{B}$
- $A\Delta B$ - $(A\cap \compl{B}) \cup (\compl{A} \cap B)$
- $\powerset(A)$ - set of all subsets of $A$
sets in metric vector spaces
- $\closure{A}$ - closure of set $A$
- $\interior{A}$ - interior of set $A$
- $\relint A$ - relative interior of set $A$
- $\boundary A$ - boundary of set $A$
set algebra
- $\sigma(\subsetset{A})$ - $\sigma$-algebra generated by $\subsetset{A}$, i.e., smallest $\sigma$-algebra containing $\subsetset{A}$
norms in $\reals^n$
- $\|x\|_p$ ($p\geq1$) - $p$-norm of $x\in\reals^n$, i.e., $(|x_1|^p + \cdots + |x_n|^p)^{1/p}$
- e.g., $\|x\|_2$ - Euclidean norm
matrices and vectors
- $a_{i}$ - $i$-th entry of vector $a$
- $A_{ij}$ - entry of matrix $A$ at position $(i,j)$, i.e., entry in $i$-th row and $j$-th column
- $\Tr(A)$ - trace of $A \in\reals^{n\times n}$, i.e., $A_{1,1}+ \cdots + A_{n,n}$
symmetric, positive definite, and positive semi-definite matrices
- $\symset{n}\subset \reals^{n\times n}$ - set of symmetric matrices
- $\possemidefset{n}\subset \symset{n}$ - set of positive semi-definite matrices; $A\succeq0 \Leftrightarrow A \in \possemidefset{n}$
- $\posdefset{n}\subset \symset{n}$ - set of positive definite matrices; $A\succ0 \Leftrightarrow A \in \posdefset{n}$
sometimes, use Python script-like notations (with serious abuse of mathematical notations)
- use $f:\reals\to\reals$ as if it were $f:\reals^n \to \reals^n$, e.g., $$ \exp(x) = (\exp(x_1), \ldots, \exp(x_n)) \quad \mbox{for } x\in\reals^n $$ and $$ \log(x) = (\log(x_1), \ldots, \log(x_n)) \quad \mbox{for } x\in\ppreals^n $$ which corresponds to Python code numpy.exp(x) or numpy.log(x) where x is instance of numpy.ndarray, i.e., numpy array
- use $\sum x$ to mean $\ones^T x$ for $x\in\reals^n$, i.e. $$ \sum x = x_1 + \cdots + x_n $$ which corresponds to Python code x.sum() where x is numpy array
- use $x/y$ for $x,y\in\reals^n$ to mean $$ \rowvecthree{x_1/y_1}{\cdots}{x_n/y_n}^T $$ which corresponds to Python code x / y where x and y are $1$-d numpy arrays
- use $X/Y$ for $X,Y\in\reals^{m\times n}$ to mean $$ \begin{my-matrix}{cccc} X_{1,1}/Y_{1,1} & X_{1,2}/Y_{1,2} & \cdots & X_{1,n}/Y_{1,n} \\ X_{2,1}/Y_{2,1} & X_{2,2}/Y_{2,2} & \cdots & X_{2,n}/Y_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ X_{m,1}/Y_{m,1} & X_{m,2}/Y_{m,2} & \cdots & X_{m,n}/Y_{m,n} \end{my-matrix} $$ which corresponds to Python code X / Y where X and Y are $2$-d numpy arrays

Some definitions

statement $P_n$, said to happen infinitely often or i.o. if $$ \left( \forall N\in\naturals \right) \left( \exists n > N \right) \left( P_n \right) $$

statement $P(x)$, said to happen almost everywhere or a.e. or almost surely or a.s. (depending on context) associated with measure space $\meas{X}{\algB}{\mu}$ if $$ \mu \set{x}{P(x)} = 1 $$ or equivalently $$ \mu \set{x}{\sim P(x)} = 0 $$

Some conventions

(for some subjects) use following conventions
- $0\cdot \infty = \infty \cdot 0 = 0$
- $(\forall x\in\ppreals)(x\cdot \infty = \infty \cdot x = \infty)$
- $\infty \cdot \infty = \infty$

Real Analysis

Set Theory

Some principles

$$ P(1) \& [P(n\Rightarrow P(n+1)] \Rightarrow (\forall n \in \naturals)P(n) $$

each nonempty subset of $\naturals$ has a smallest element

for $f:X\to X$ and $a\in X$, exists unique infinite sequence $\langle x_n\rangle_{n=1}^\infty\subset X$ such that $$ x_1=a $$ and $$ \left( \forall n \in \naturals \right) \left( x_{n+1} = f(x_n) \right) $$

note that $\Leftrightarrow$ $\Rightarrow$

Some definitions for functions

for $f:X\to Y$

terms, map and function, exterchangeably used
$X$ and $Y$, called domain of $f$ and codomain of $f$ respectively
$\set{f(x)}{x\in X}$, called range of $f$
for $Z\subset Y$, $f^{-1}(Z) = \set{x\in X}{f(x)\in Z}\subset X$, called preimage or inverse image of $Z$ under $f$
for $y\in Y$, $f^{-1}(\{y\})$, called fiber of $f$ over $y$
$f$, called injective or injection or one-to-one if $\left( \forall x\neq v \in X \right) \left( f(x) \neq f(v) \right)$
$f$, called surjective or surjection or onto if $\left( \forall x \in X \right) \left( \exists y in Y \right) (y=f(x))$
$f$, called bijective or bijection if $f$ is both injective and surjective, in which case, $X$ and $Y$, said to be one-to-one correspondece or bijective correspondece
$g:Y\to X$, called left inverse if $g\circ f$ is identity function
$h:Y\to X$, called right inverse if $f\circ h$ is identity function

Some properties of functions

for $f:X\to Y$

$f$ is injective if and only if $f$ has left inverse
$f$ is surjective if and only if $f$ has right inverse
hence, $f$ is bijective if and only if $f$ has both left and right inverse because if $g$ and $h$ are left and right inverses respectively, $g = g \circ (f\circ h) = (g\circ f)\circ h = h$
if $|X|=|Y|<\infty$, $f$ is injective if and only if $f$ is surjective if and only if $f$ is bijective

Countability of sets

set $A$ is countable if range of some function whose domain is $\naturals$
$\naturals$, $\integers$, $\rationals$: countable
$\reals$: not countable

Limit sets

for sequence, $\seq{A_n}$, of subsets of $X$
- limit superior or limsup of \seq{A_n}, defined by $$ \limsup \seq{A_n} = \bigcap_{n=1}^\infty \bigcup_{m=n}^\infty A_m $$
- limit inferior or liminf of \seq{A_n}, defined by $$ \liminf \seq{A_n} = \bigcup_{n=1}^\infty \bigcap_{m=n}^\infty A_m $$
always $$ \liminf \seq{A_n} \subset \limsup \seq{A_n} $$
when $\liminf \seq{A_n} = \limsup \seq{A_n}$, sequence, $\seq{A_n}$, said to converge to it, denote $$ \lim \seq{A_n} = \liminf \seq{A_n} = \limsup \seq{A_n} = A $$

Algebras of sets

collection $\alg$ of subsets of $X$ called algebra or Boolean algebra if $$ (\forall A, B \in \alg) (A\cup B\in\alg) \mbox{ and } (\forall A \in \alg) (\compl{A}\in\alg) $$
- $(\forall A_1, \ldots, A_n \in \alg)(\cup_{i=1}^n A_i \in \alg)$
- $(\forall A_1, \ldots, A_n \in \alg)(\cap_{i=1}^n A_i \in \alg)$
algebra $\alg$ called $\sigma$-algebra or Borel field if
- every union of a countable collection of sets in $\alg$ is in $\alg$, i.e., $$ (\forall \seq{A_i})(\cup_{i=1}^\infty A_i \in \alg) $$
given sequence of sets in algebra $\alg$, $\seq{A_i}$, exists disjoint sequence, $\seq{B_i}$ such that $$ B_i \subset A_i \mbox{ and } \bigcup_{i=1}^\infty B_i = \bigcup_{i=1}^\infty A_i $$

Algebras generated by subsets

algebra generated by collection of subsets of $X$, $\coll$, can be found by $$ \alg = \bigcap \set{\algk{B}}{\algk{B} \in \collF} $$ where $\collF$ is family of all algebras containing $\coll$
- smallest algebra $\alg$ containing $\coll$, i.e., $$ (\forall \algk{B} \in \collF)(\alg \subset \algk{B}) $$
$\sigma$-algebra generated by collection of subsets of $X$, $\coll$, can be found by $$ \alg= \bigcap \set{\algk{B}}{\algk{B} \in \collG} $$ where $\collG$ is family of all $\sigma$-algebras containing $\coll$
- smallest $\sigma$-algebra $\alg$ containing $\coll$, i.e., $$ (\forall \algk{B} \in \collG)(\alg \subset \algk{B}) $$

Relation

$x$ said to stand in relation $\rel$ to $y$, denoted by $\relxy{x}{y}$
$\rel$ said to be relation on $X$ if $\relxy{x}{y}$ $\Rightarrow$ $x\in X$ and $y\in X$
$\rel$ is
- transitive if $\relxy{x}{y}$ and $\relxy{y}{z}$ $\Rightarrow$ $\relxy{x}{z}$
- symmetric if $\relxy{x}{y} = \relxy{y}{x}$
- reflexive if $\relxy{x}{x}$
- antisymmetric if $\relxy{x}{y}$ and $\relxy{y}{x}$ $\Rightarrow$ $x=y$
$\rel$ is
- equivalence relation if transitive, symmetric, and reflexive, e.g., modulo
- partial ordering if transitive and antisymmetric, e.g., “$\subset$''
- linear (or simple) ordering if transitive, antisymmetric, and $\relxy{x}{y}$ or $\relxy{y}{x}$ for all $x,y\in X$
  - e.g., “$\geq$'' linearly orders $\reals$ while “$\subset$'' does not $\powerset(X)$

Ordering

given partial order, $\prec$, $a$ is
- a first/smallest/least element if $x \neq a \Rightarrow a\prec x$
- a last/largest/greatest element if $x \neq a \Rightarrow x\prec a$
- a minimal element if $x \neq a \Rightarrow x \not\prec a$
- a maximal element if $x \neq a \Rightarrow a \not\prec x$
partial ordering $\prec$ is
- strict partial ordering if $x\not\prec x$
- reflexive partial ordering if $x\prec x$
strict linear ordering $<$ is
- well ordering for $X$ if every nonempty set contains a first element

Axiom of choice and equivalent principles

given a collection of nonempty sets, $\coll$, there exists $f:\coll\ \to \cup_{A\in\coll} A$ such that $$ \left( \forall A\in\coll\ \right) \left( f(A) \in A \right) $$

also called multiplicative axiom - preferred to be called to axiom of choice by Bertrand Russell for reason writte
no problem when $\coll$ is finite
need axiom of choice when $\coll$ is not finite

for particial ordering $\prec$ on $X$, exists a maximal linearly ordered subset $S\subset X$, i.e., $S$ is linearity ordered by $\prec$ and if $S\subset T\subset X$ and $T$ is linearly ordered by $\prec$, $S=T$

every set $X$ can be well ordered, i.e., there is a relation $<$ that well orders $X$

note that $\Leftrightarrow$ $\Leftrightarrow$

Infinite direct product

for collection of sets, $\seq{X_\lambda}$, with index set, $\Lambda$, $$ \bigtimes_{\lambda\in\Lambda} X_\lambda $$ called direct product

for $z=\seq{x_\lambda}\in\bigtimes X_\lambda$, $x_\lambda$ called $\lambda$-th coordinate of $z$

if one of $X_\lambda$ is empty, $\bigtimes X_\lambda$ is empty
axiom of choice is equivalent to converse, i.e., if none of $X_\lambda$ is empty, $\bigtimes X_\lambda$ is not empty if one of $X_\lambda$ is empty, $\bigtimes X_\lambda$ is empty
this is why Bertrand Russell prefers multiplicative axiom to axiom of choice for name of axiom ()

Real Number System

Field axioms

field axioms - for every $x,y,z\in\field$
- $(x+y)+z= x+(y+z)$ - additive associativity
- $(\exists 0\in\field)(\forall x\in\field)(x+0=x)$ - additive identity
- $(\forall x\in\field)(\exists w\in\field)(x+w=0)$ - additive inverse
- $x+y= y+x$ - additive commutativity
- $(xy)z= x(yz)$ - multiplicative associativity
- $(\exists 1\neq0\in\field)(\forall x\in\field)(x\cdot 1=x)$ - multiplicative identity
- $(\forall x\neq0\in\field)(\exists w\in\field)(xw=1)$ - multiplicative inverse
- $x(y+z) = xy + xz$ - distributivity
- $xy= yx$ - multiplicative commutativity
system (set with $+$ and $\cdot$) satisfying axiom of field called field
- e.g., field of module $p$ where $p$ is prime, $\primefield{p}$

Axioms of order

axioms of order - subset, $\field_{++}\subset \field$, of positive (real) numbers satisfies
- $x,y\in \field_{++} \Rightarrow x+y\in \field_{++}$
- $x,y\in \field_{++} \Rightarrow xy\in \field_{++}$
- $x\in \field_{++} \Rightarrow -x\not\in \field_{++}$
- $x\in \field \Rightarrow x=0\lor x\in \field_{++} \lor -x \in \field_{++}$
system satisfying field axioms & axioms of order called ordered field
- e.g., set of real numbers ($\reals$), set of rational numbers ($\rationals$)

Axiom of completeness

completeness axiom
- every nonempty set $S$ of real numbers which has an upper bound has a least upper bound, i.e., $$ \set{l}{(\forall x\in S)(l\leq x)} $$ has least element.
- use $\inf S$ and $\sup S$ for least and greatest element (when exist)
ordered field that is complete is complete ordered field
- e.g., $\reals$ (with $+$ and $\cdot$)
axiom of Archimedes
- given any $x\in\reals$, there is an integer $n$ such that $x<n$
corollary
- given any $x<y \in \reals$, exists $r\in\rationals$ such tat $x < r < y$

Sequences of $\reals$

sequence of $\reals$ denoted by $\seq{x_i}_{i=1}^\infty$ or $\seq{x_i}$
- mapping from $\naturals$ to $\reals$
limit of $\seq{x_n}$ denoted by $\lim_{n\to\infty} x_n$ or $\lim x_n$ - defined by $a\in\reals$ $$ (\forall \epsilon>0)(\exists N\in\naturals) (n \geq N \Rightarrow |x_n-a|<\epsilon) $$
- $\lim x_n$ unique if exists
$\seq{x_n}$ called Cauchy sequence if $$ (\forall \epsilon>0)(\exists N\in\naturals) (n,m \geq N \Rightarrow |x_n-x_m|<\epsilon) $$
Cauchy criterion - characterizing complete metric space (including $\reals$)
- sequence converges if and only if Cauchy sequence

Other limits

cluster point of $\seq{x_n}$ - defined by $c\in\reals$ $$ (\forall \epsilon>0, N\in\naturals)(\exists n>N)(|x_n-c|<\epsilon) $$
limit superior or limsup of $\seq{x_n}$ $$ \limsup x_n = \inf_n \sup_{k>n} x_k $$
limit inferior or liminf of $\seq{x_n}$ $$ \liminf x_n = \sup_n \inf_{k>n} x_k $$
$\liminf x_n \leq \limsup x_n$
$\seq{x_n}$ converges if and only if $\liminf x_n = \limsup x_n$ (=$\lim x_n$)

Open and closed sets

$O$ called open if $$ (\forall x\in O)(\exists \delta>0)(y\in\reals)(|y-x|<\delta\Rightarrow y\in O) $$
- intersection of finite collection of open sets is open
- union of any collection of open sets is open
$\closure{E}$ called closure of $E$ if $$ (\forall x \in \closure{E} \ \&\ \delta>0)(\exists y\in E)(|x-y|<\delta) $$
$F$ called closed if $$ F = \closure{F} $$
- union of finite collection of closed sets is closed
- intersection of any collection of closed sets is closed

Open and closed sets - facts

every open set is union of countable collection of disjoint open intervals
(Lindelöf) any collection $\coll$ of open sets has a countable subcollection $\seq{O_i}$ such that $$ \bigcup_{O\in\coll} O = \bigcup_{i} O_i $$
- equivalently, any collection $\collk{F}$ of closed sets has a countable subcollection $\seq{F_i}$ such that $$ \bigcap_{O\in\collk{F}} F = \bigcap_{i} F_i $$

Covering and Heine-Borel theorem

collection $\coll$ of sets called covering of $A$ if $$ A \subset \bigcup_{O\in\coll} O $$
- $\coll$ said to cover $A$
- $\coll$ called open covering if every $O\in\coll$ is open
- $\coll$ called finite covering if $\coll$ is finite
Heine-Borel theorem\index{Heine-Borel theorem}\index{Heine, Heinrich Eduard!Heine-Borel theorem}\index{Borel, Félix Édouard Justin Émile!Heine-Borel theorem} - for any closed and bounded set, every open covering has finite subcovering
corollary
- any collection $\coll$ of closed sets including at least one bounded set every finite subcollection of which has nonempty intersection has nonempty intersection.

Continuous functions

$f$ (with domain $D$) called continuous at $x$ if $$ (\forall\epsilon >0)(\exists \delta>0)(\forall y\in D)(|y-x|<\delta \Rightarrow |f(y)-f(x)|<\epsilon) $$
$f$ called continuous on $A\subset D$ if $f$ is continuous at every point in $A$
$f$ called uniformly continuous on $A\subset D$ if $$ (\forall\epsilon >0)(\exists \delta>0)(\forall x,y\in D)(|x-y|<\delta \Rightarrow |f(x)-f(y)|<\epsilon) $$

Continuous functions - facts

$f$ is continuous if and only if for every open set $O$ (in co-domain), $f^{-1}(O)$ is open
$f$ continuous on closed and bounded set is uniformly continuous
extreme value theorem - $f$ continuous on closed and bounded set, $F$, is bounded on $F$ and assumes its maximum and minimum on $F$ $$ (\exists x_1, x_2 \in F)(\forall x\in F)(f(x_1) \leq f(x) \leq f(x_2)) $$
intermediate value theorem - for $f$ continuous on $[a,b]$ with $f(a) \leq f(b)$, $$ (\forall d)(f(a) \leq d \leq f(b))(\exists c\in[a,b])(f(c) = d) $$

Borel sets and Borel $\sigma$-algebra

Borel set
- any set that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement
Borel algebra or Borel $\sigma$-algebra
- smallest $\sigma$-algebra containing all open sets
- also
  - smallest $\sigma$-algebra containing all closed sets
  - smallest $\sigma$-algebra containing all open intervals (due to statement on page~)

Various Borel sets

countable union of closed sets (in $\reals$), called an $F_\sigma$ ($F$ for closed & $\sigma$ for sum)
- thus, every countable set, every closed set, every open interval, every open sets, is an $F_\sigma$ (note $(a,b)=\bigcup_{n=1}^\infty [a+1/n,b-1/n]$)
- countable union of sets in $F_\sigma$ again is an $F_\sigma$
countable intersection of open sets called a $G_\delta$ ($G$ for open & $\delta$ for durchschnitt - average in German)
- complement of $F_\sigma$ is a $G_\delta$ and vice versa
$F_\sigma$ and $G_\delta$ are simple types of Borel sets
countable intersection of $F_\sigma$'s is $F_{\sigma\delta}$, countable union of $F_{\sigma\delta}$'s is $F_{\sigma\delta\sigma}$, countable intersection of $F_{\sigma\delta\sigma}$'s is $F_{\sigma\delta\sigma\delta}$, etc., & likewise for $G_{\delta \sigma \ldots}$
below are all classes of Borel sets, but not every Borel set belongs to one of these classes $$ F_{\sigma}, F_{\sigma\delta}, F_{\sigma\delta\sigma}, F_{\sigma\delta\sigma\delta}, \ldots, G_{\delta}, G_{\delta\sigma}, G_{\delta\sigma\delta}, G_{\delta\sigma\delta\sigma}, \ldots, $$

Space Overview

Diagrams for relations among various spaces

note from the figure
- metric should be defined to utter completeness
- metric spaces can be induced from normed spaces

Classical Banach Spaces

Normed linear space

$X$ called linear space if $$ (\forall x, y \in X, a, b \in \reals)(ax + by \in X) $$
linear space, $X$, called normed space with associated norm $\|\cdot\|: X \to \preals$ if
- $$ (\forall x\in X)(\|x\|=0 \Rightarrow x \equiv 0) $$
- $$ (\forall x \in X, a \in \reals)(\|ax\| = |a|\|x\|) $$
- subadditivity $$ (\forall x,y\in X)(\|x+y\| \leq \|x\| + \|y\|) $$

$L^p$ spaces

$L^p = L^p[0,1]$ denotes space of (Lebesgue) measurable functions such that $$ \int_{[0,1]} |f|^p < \infty $$
define $\|\cdot\|:L^p\to\preals$ $$ \|f\| = \|f\|_p = \left(\int_{[0,1]} |f|^p\right)^{1/p} $$
$L^p$ are linear normed spaces with norm $\|\cdot\|_p$ when $p\geq 1$ because
- $|f(x)|^p + |g(x)|^p \leq 2^p(|f(x)|^p + |g(x)|^p)$ implies $(\forall f, g\in L^p)(f+g \in L^p)$
- $|\alpha f(x)|^p = |a|^p|f(x)|^p$ implies $(\forall f\in L^p, a \in \reals)(af \in L^p)$
- $\|f\|=0\Rightarrow f=0\mbox{ a.e.}$
- $\|a f\| = |a|\|f\|$
- $\|f+g\|\geq \|f\|+\|g\|$ (Minkowski inequality)

$L^\infty$ space

$L^\infty = L^\infty[0,1]$ denotes space of measurable functions bounded a.e.
$L^\infty$ is linear normed space with norm $$ \|f\| = \|f\|_\infty = \mathrm{ess\ sup} |f| = \inf_{g: g=f \ \mathrm{a.e}} \sup_{x\in[0,1]} |g(x)| $$
- thus $$ \|f\|_\infty = \inf\set{M}{\mu\set{x}{f(x)>M}=0} $$

Inequalities in $L^\infty$

Minkowski inequality - for $p\in[1,\infty]$ $$ (\forall f,g\in L^p)(\|f+g\|_p \leq \|f\|_p + \|g\|_p) $$
- if $p\in(1,\infty)$, equality holds if and only if $(\exists a,b\geq 0 \mbox{ with } ab\neq0)(af = bg \mbox{ a.e.})$
Minkowski inequality for $0<p<1$: $$ (\forall f,g\in L^p)(f,g\geq0 \mbox{ a.e.} \Rightarrow \|f+g\|_p \geq \|f\|_p + \|g\|_p) $$
Hölder's inequality - for $p,q\in[1,\infty]$ with $1/p+1/q=1$ $$ (\forall f\in L^p, g\in L^q) \left(fg \in L^1 \mbox{ and } \int_{[0,1]} |fg| \leq \int_{[0,1]} |f|^p \int_{[0,1]} |g|^q\right) $$
- equality holds if and only if $(\exists a,b\geq 0 \mbox{ with } ab\neq0)(a|f|^p = b|g|^q \mbox{ a.e.})$

Convergence and completeness in normed linear spaces

$\seq{f_n}$ in normed linear space
- said to converge to $f$, i.e., $\lim f_n =f$ or $f_n \to f$, if $$ (\forall \epsilon>0)(\exists N\in\naturals)(\forall n> N)(\|f_n-f\|<\epsilon) $$
- called Cauchy sequence if $$ (\forall \epsilon>0)(\exists N\in\naturals)(\forall n,m> N)(\|f_n-f_m\|<\epsilon) $$
- called summable if $\sum^n_{i=1} f_i$ converges
- called absolutely summable if $\sum^n_{i=1} |f_i|$ converges
normed linear space called complete if every Cauchy sequence converges
normed linear space is complete if and only if every absolutely summable series is summable

Banach space

complete normed linear space called Banach space
(Riesz-Fischer) $L^p$ spaces are compact, hence Banach spaces
convergence in $L^p$ called convergence in mean of order $p$
convergence in $L^\infty$ implies nearly uniformly converges

Approximation in $L^p$

$\Delta=\seq{d_i}_{i=0}^n$ with $0=d_1<d_2<\cdots<d_n=1$ called subdivision of $[0,1]$ (with $\Delta_i = [d_{i-1},d_{i}]$)
$\varphi_{f,\Delta}$ for $f\in L^p$ called step function if $$ \varphi_{f,\Delta}(x) = \frac{1}{d_i-d_{i+1}}\int_{d_{i-1}}^{d_i} f(t)dt \mbox{ for } x\in[d_{i-1},d_i) $$
for $f\in L^p$ ($1<p\leq \infty$), exist $\varphi_{f,\Delta}$ and continuous function, $\psi$ such that $$ \|\varphi_{f,\Delta_i}-f\|<\epsilon \mbox{ and } \|\psi-f\|<\epsilon $$
- $L^p$ version of Littlewood's second principle (page~)
for $f\in L^p$, $\varphi_{f,\Delta}\to f$ as $\max \Delta_i\to0$, i.e., $$ (\forall \epsilon>0)(\exists \delta>0)(\max \Delta_i < \delta \Rightarrow \|\varphi_{f,\Delta}-f\|_p < \epsilon) $$

Bounded linear functionals on $L^p$

$F:X\in\reals$ for normed linear space $X$ called linear functional if $$ (\forall f, g \in F, a,b \in\reals)(F(af+bg)=aF(f)+bF(g)) $$
linear functional, $F$, said to be bounded if $$ (\exists M)(\forall f\in X)(|F(f)|\leq M\|f\|) $$
smallest such constant called norm of $F$, i.e., $$ \|F\| = \sup_{f\in X, f\neq0} {|F(f)|}/{\|f\|} $$

Riesz representation theorem

for every $g\in L^q$ ($1\leq p\leq \infty$), following defines a bounded linear functional in $L^p$ $$ F(f) = \int fg $$ where $\|F\|=\|g\|_q$
Riesz representation theorem - for every bounded linear functional in $L^p$, $F$, ($1\leq p<\infty$), there exists $g\in L^q$ such that $$ F(f) = \int fg $$ where $\|F\|=\|g\|_q$
for each case, $L^q$ is dual of $L^p$ (refer to page for definition of dual)

Metric Spaces

Metric spaces

$\metrics{X}{\rho}$ with nonempty set, $X$, and metric $\rho: X\times X\to\preals$ called metric space if for every $x,y,z \in X$
- $\rho(x,y)=0 \Leftrightarrow x=y$
- $\rho(x,y)=\rho(y,x)$
- $\rho(x,y) \leq \rho(x,z) + \rho(z,y)$ (triangle inequality)
examples of metric spaces
- $\metrics{\reals}{|\cdot|}$, $\metrics{\reals^n}{\|\cdot\|_p}$ with $1\leq p\leq \infty$
for $f\subset X$, $S_{x,r} = \set{y}{\rho(y,x)<r}$ called ball
for $E\subset X$, $\sup\set{\rho(x,y)}{x,y \in E}$ called diameter of $E$ defined by
$\rho$ called pseudometric if 1st requirement removed
$\rho$ called extended metric if $\rho: X\times X \to\preals\cup\{\infty\}$

Cartesian product

for two metric spaces $\metrics{X}{\rho}$ and $\metrics{Y}{\sigma}$, metric space $\metrics{X\times Y}{\tau}$ with $\tau:X\times Y\to\preals$ such that $$ \tau((x_1,y_1),(x_2,y_2)) = (\rho(x_1,x_2)^2 + \sigma(y_1,y_2)^2)^{1/2} $$ called Cartesian product metric space
$\tau$ satisfies all properties required by metric
- e.g., $\reals^{n} \times \reals^{m} = \reals^{n+m}$

Open sets - metric spaces

$O \subset X$ said to be open open if $$ (\forall x\in O)(\exists \delta>0)(\forall y\in X)(\rho(y,x)<\delta \Rightarrow y\in O) $$
- $X$ and $\emptyset$ are open
- intersection of finite collection of open sets is open
- union of any collection of open sets is open

Closed sets - metric spaces

$x\in X$ called point of closure of $E\subset X$ if $$ (\forall \epsilon>0)(\exists y\in E)(\rho(y,x) < \epsilon) $$
- $\closure{E}$ denotes set of points of closure of $E$; called closure of $E$
- $E\subset \closure{E}$
$F \subset X$ said to be closed if $$ F = \closure{F} $$
- $X$ and $\emptyset$ are closed
- union of finite collection of closed sets is closed
- intersection of any collection of closed sets is closed
complement of closed set is open
complement of open set is closed

Dense sets and separability - metric spaces

$D\subset X$ said to be dense if $$ \closure{D} = X $$
$X$ is said to be separable if exists finite dense subset, i.e., $$ (\exists D\subset X)(|D| < \infty \ \& \ \closure{D}=X) $$
$X$ is separable if and only if exists countable collection of open sets $\seq{O_i}$ such that for all open $O\subset X$ $$ O = \bigcup_{O_i\subset O} O_i $$

Continuous functions - metric spaces

$f:X\to Y$ for metric spaces $\metrics{X}{\rho}$ and $\metrics{Y}{\sigma}$ called mapping or function from $X$ into $Y$
$f$ said to be onto if $$ f(X)=Y $$
$f$ said to be continuous at $x\in X$ if $$ (\forall \epsilon>0)(\exists \delta>0)(\forall y\in X)(\rho(y,x)<\delta \Rightarrow \sigma(f(y),f(x))<\epsilon) $$
$f$ said to be continuous if $f$ is continuous at every $x\in X$
$f$ is continuous if and only if for every open $O\subset Y$, $f^{-1}(O)$ is open
if $f:X\to Y$ and $g:Y\to Z$ are continuous, $g\circ f:X\to Z$ is continuous

Homeomorphism

one-to-one mapping of $X$ onto $Y$ (or equivalently, one-to-one correspondece between $X$ and $Y$), $f$, said to be homeomorphism if
- both $f$ and $f^{-1}$ are continuous
$X$ and $Y$ said to be homeomorphic if exists homeomorphism
topology is study of properties unaltered by homeomorphisms and such properties called topological
one-to-one correspondece $X$ and $Y$ is homeomorphism if and only if it maps open sets in $X$ to open sets in $Y$ and vice versa
every property defined by means of open sets (or equivalently, closed sets) or/and being continuous functions is topological one
- e.g., $f$ is continuous on $X$ is homeomorphism, then $f\circ h^{-1}$ is continuous function on $Y$

Isometry

homeomorphism preserving distance called isometry, i.e., $$ (\forall x,y \in X)(\sigma(h(x),h(y)) = \rho(x,y)) $$
$X$ and $Y$ said to be isometric if exists isometry
(from abstract point of view) two isometric spaces are exactly same; it's nothing but relabeling of points
two metrics, $\rho$ and $\sigma$ on $X$, said to be equivalent if identity mapping of $\metrics{X}{\rho}$ onto $\metrics{X}{\sigma}$ is homeomorphism
- hence, two metrics are equivalent if and only if set in one metric is open whenever open in the other metric

Convergence - metric spaces

$\seq{x_n}$ defined for metric space, $X$
- said to converge to $x$, i.e., $\lim x_n =x$ or $x_n \to x$, if $$ (\forall \epsilon>0)(\exists N\in\naturals)(\forall n> N)(\rho(x_n,x)<\epsilon) $$
  - equivalently, every ball about $x$ contains all but finitely many points of $\seq{x_n}$
- said to have cluster point, $x$, if $$ (\forall \epsilon>0, N\in\naturals)(\exists n> N)(\rho(x_n,x)<\epsilon) $$
  - equivalently, every ball about $x$ contains infinitely many points of $\seq{x_n}$
  - equivalently, every ball about $x$ contains at least one point of $\seq{x_n}$
every convergent point is cluster point
- converse not true

Completeness - metric spaces

$\seq{x_n}$ of metric space, $X$, called Cauchy sequence if $$ (\forall \epsilon>0)(\exists N\in\naturals)(\forall n,m> N)(\rho(x_n,x_m)<\epsilon) $$
convergence sequence is Cauchy sequence
$X$ said to be complete if every Cauchy sequence converges
- e.g., $\metrics{\reals}{\rho}$ with $\rho(x,y)=|x-y|$
for incomplete $\metrics{X}{\rho}$, exists complete $X^\ast$ where $X$ is isometrically embedded in $X^\ast$ as dense set
if $X$ contained in complete $Y$, $X^\ast$ is isometric with $\closure{X}$ in $Y$

Uniform continuity - metric spaces

$f:X\to Y$ for metric spaces $\metrics{X}{\rho}$ and $\metrics{Y}{\sigma}$ said to be uniformly continuous if $$ (\forall \epsilon>0)(\exists \delta)(\forall x,y \in X)(\rho(x,y) < \delta \Rightarrow \sigma(f(x),f(y))<\epsilon) $$
- example of continuous, but not uniformly continuous function
  - $h:[0,1)\to\preals$ with $h(x)=x/(1-x)$
  - $h$ maps Cauchy sequence $\seq{1-1/n}_{n=1}^\infty$ in $[0,1)$ to $\seq{n-1}_{n=1}^\infty$ in $\preals$, which is not Cauchy sequence
homeomorphism $f$ between $\metrics{X}{\rho}$ and $\metrics{Y}{\sigma}$ with both $f$ and $f^{-1}$ uniformly continuous called uniform homeomorphism

Uniform homeomorphism

uniform homeomorphism $f$ between $\metrics{X}{\rho}$ and $\metrics{Y}{\sigma}$ maps every Cauchy sequence $\seq{x_n}$ in $X$ mapped to $\seq{f(x_n)}$ in $Y$ which is Cauchy
- being Cauchy sequence, hence, being complete preserved by uniform homeomorphism
- being uniformly continuous also preserved by uniform homeomorphism
each of three properties (being Cauchy sequence, being complete, being uniformly continuous) called uniform property
uniform properties are not topological properties, e.g., $h$ on page~
- is homeomorphism between incomplete space $[0,1)$ and complete space $\preals$
- maps Cauchy sequence $\seq{1-1/n}_{n=1}^\infty$ in $[0,1)$ to $\seq{n-1}_{n=1}^\infty$ in $\preals$, which is not Cauchy sequence
- its inverse maps uniformly continuous function $\sin$ back to non-uniformly continuity function on $[0,1)$

Uniform equivalence

two metrics, $\rho$ and $\sigma$ on $X$, said to be uniformly equivalent if identity mapping of $\metrics{X}{\rho}$ onto $\metrics{X}{\sigma}$ is uniform homeomorphism, i.e., $$ (\forall \epsilon, \delta>0, x,y \in X) (\rho(x,y)<\delta \Rightarrow \sigma(x,y)<\epsilon \ \&\ \sigma(x,y)<\delta \Rightarrow \rho(x,y)<\epsilon) $$
example of uniform equivalence on $X\times Y$
- any two of below metrics are uniformly equivalent on $X\times Y$ $$ \begin{eqnarray*} &&\tau((x_1,y_1),(x_2,y_2)) = (\rho(x_1,x_2)^2 + \sigma(y_1,y_2)^2)^{1/2} \\ &&\rho_1((x_1,y_1),(x_2,y_2)) = \rho(x_1,x_2) + \sigma(y_1,y_2) \\ &&\rho_\infty((x_1,y_1),(x_2,y_2)) = \max\{\rho(x_1,x_2), \sigma(y_1,y_2)\} \end{eqnarray*} $$
for $\metrics{X}{\rho}$ and complete $\metrics{Y}{\sigma}$ and $f:X\to Y$ uniformly continuous on $E\subset X$ into $Y$, exists unique continuous extension $g$ of $f$ on $\closure{E}$, which is uniformly continuous

Subspaces

for metric space, $\metrics{X}{\rho}$, metric space $\metrics{S}{\rho_S}$ with $S\subset X$ and $\rho_S$ being restriction of $\rho$ to S, called subspace of $\metrics{X}{\rho}$
- e.g. (with standard Euclidean distance)
  - $\rationals$ is subspace of $\reals$
  - $\bigsetl{(x,y)\in\reals^2}{y=0}$ is subspace of $\reals^2$, which is isometric to $\reals$
for metric space, $X$, and its subspace, $S$,
- $\closure{E}\subset S$ is closure of $E$ relative to $S$.
- $A\subset S$ is closure relative to $S$ if and only if $(\exists \closure{F}\subset A)(A = \closure{F}\cap S)$
- $A\subset O$ is open relative to $S$ if and only if $(\exists \mbox{ open }{O}\subset A)(A = {O}\cap S)$
also
- every subspace of separable metric space is separable
- every complete subset of metric space is closed
- every closed subset of complete metric space is complete

Compact metric spaces

motivation - want metric spaces where
- conclusion of Heine-Borel theorem (page~) are valid
- many properties of $[0,1]$ are true, e.g., Bolzano-Weierstrass property (page~)
e.g.,
- bounded closed set in $\reals$ has finite open covering property
metric space $X$ called compact metric space if every open covering of $X$, $\collk{U}$, contains finite open covering of $X$, e.g., $$ (\forall \mbox{ open covering of $X$}, \collk{U})(\exists \{O_1,\ldots,O_n\} \subset \collk{U}) (X\in\cup O_i) $$
$A\subset X$ called compact if compact as subspace of $X$
- i.e., every open covering of $A$ contains finite open covering of $A$

Compact metric spaces - alternative definition

collection, $\collk{F}$, of sets in $X$ said to have finite intersection property if every finite subcollection of $\collk{F}$ has nonempty intersection
if rephrase definition of compact metric spaces in terms of closed instead of open
- $X$ is called compact metric space if every collection of closed sets with empty intersection contains finite subcollection with empty intersection
thus, $X$ is compact if and only if every collection of closed sets with finite intersection property has nonempty intersection

Bolzano-Weierstrass property and sequential compactness

metric space said to
- have Bolzano-Weierstrass property if every sequence has cluster point
- $X$ said to be sequentially compact if every sequence has convergent subsequence
$X$ has Bolzano-Weierstrass property \iaoi\ sequentially compact

Compact metric spaces - properties

following three statements about metric space are equivalent (not true for general topological sets)
- being compact
- having Bolzano-Weierstrass property
- being sequentially compact
compact metric spaces have corresponding to some of those of complete metric spaces (compare with statements on page~)
- every compact subset of metric space is closed and bounded
- every closed subset of compact metric space is compact
(will show above in following slides)

Necessary condition for compactness

compact metric space is sequentially compact
equivalently, compact metric space has Bolzano-Weierstrass property (page~)

Necessary conditions for sequentially compactness

every continuity real-valued function on sequentially compact space is bounded and assumes its maximum and minimum
sequentially compact space is totally bounded
every open covering of sequentially compact space has Lebesgue number

Sufficient conditions for compactness

metric space that is totally bounded and has Lebesgue number for every covering is compact

Borel-Lebesgue theorem

conditions on pages , , and imply the following equivalent statements
- $X$ is compact
- $X$ has Bolzano-Weierstrass property
- $X$ is sequentially compact
above called Borel-Lebesgue theorem
hence, can drop sequentially in every statement on page~, i.e.,
- every continuity real-valued function on ~~sequentially~~ compact space is bounded and assumes its maximum and minimum
- ~~sequentially~~ compact space is totally bounded
- every open covering of ~~sequentially~~ compact space has Lebesgue number

Compact metric spaces - other facts

closed subset of compact space is compact
compact subset of metric space is closed and bounded
- hence, Heine-Borel theorem (page~) implies
- set of $\reals$ is compact \iaoi\ closed and bounded
metric space is compact if and only if it is complete and totally bounded
thus, compactness can be viewed as absolute type of closedness
- refer to page~ for exactly same comments for general topological spaces
continuous image of compact set is compact
continuous mapping of compact metric space into metric space is uniformly continuous

Diagrams for relations among metric spaces

the figure shows relations among metric spaces stated on pages , , , and

Baire category

do (more) deeply into certain aspects of complete metric spaces, namely, Baire theory of category
subset $E$ in metric space where $\sim (\closure{E})$ is dense, said to be nowhere dense
- equivalently, $\closure{E}$ contains no nonempty open set
union of countable collection of nowhere open sets, said to be of first category or meager
set not of first category, said to be of second category or nonmeager
complement of set of first category, called residual or co-meager

Baire category theorem

Baire theorem - for complete metric space, $X$, and countable collection of dense open subsets, $\seq{O_k}\subset X$, the intersection of the collection $$ \bigcap O_k $$ is dense
- refer to page~ for locally compact space version of Baire theorem
Baire category theorem - no nonempty open subset of complete metric space is of first category, i.e., union of countable collection of nowhere dense subsets
Baire category theorem is unusual in that uniform property, \ie, completeness of metric spaces, implies purely topological nature

Second category everywhere

metric or topological spaces with property that “no nonempty open subset of complete metric space is of first category'', said to be of second category everywhere (with respect to themselves)
Baire category theorem says complete metric space is of second category everywhere
locally compact Hausdorff spaces are of second category everywhere, too (refer to page~ for definition of locally compact Hausdorff spaces)
- for these spaces, though, many of results of category theory follow directly from local compactness

Sets of first category

collection of sets with following properties, called a $\sigma$-ideal of sets
- countable union of sets in the collection is, again, in the collection
- subset of any in the collection is, again, in the collection
both of below collections are $\sigma$-ideal of sets
- sets of first category in topological space
- measure zero sets in complete measure space
sets of first category regards as “small'' sets
- such sets in complete metric spaces no interior points
interestingly! set of first category in $[0,1]$ can have Lebesgue measure $1$, hence complement of which is residual set of measure zero

Some facts of category theory

for open set, $O$, and closed set, $F$, $\closure{O}\sim O$ and $F\sim \interior{F}$ are nowhere dense
closed set of first category in complete metric space is nowhere dense
subset of complete metric space is residual if and only if contains dense $G_\delta$, hence subset of complete metric space is of first category if and only if contained in $F_\sigma$ whose complement is dense
for countable collection of closed sets, $\seq{F_n}$, $\bigcup \interior{F_n}$ is residual open set; if $\bigcup F_n$ is complete metric space, $O$ is dense
some applications of category theory to analysis seem almost too good to be belived; here's one:
uniform boundedness principle - for family, $\collF$, of real-valued continuous functions on complete metric space, $X$, with property that $(\forall x\in X)(\exists M_x\in\reals)(\forall f\in\collF)(|f(x)|\leq M_x)$ $$ (\exists \mbox{ open }O, M\in\reals)(\forall x\in O, f\in\collF)(|f(x)|\leq M) $$