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 47: Osmosis topics

Sets, Functions, Relations, Posets, $ℝ\text{.}$

Sets

elements, empty set, subset, union, intersection, disjoint, product of sets

Functions

injective, surjective, bijective, equal functions, inverse function, restriction, identity function, composition of functions.

(Important Theorem) Let $f:S\to T$ be a function. An inverse function to $f$ exists if and only if $f$ is bijective.

Cardinality: isomorphism of sets

finite, infinite, countable, uncountable.

HW: Show that $\text{Card}\left(\mathbb{Q}\right)=\text{Card}\left(\mathbb{Z}\right)=\text{Card}\left({\mathbb{Z}}_{>0}\right)\ne \text{Card}\left(ℝ\right)\text{.}$

HW: Show that $\text{Card}\left(ℝ\right)=\text{Card}\left({ℝ}^2\right)\text{.}$

Relations

Let $S$ be a set. A relation on $S$ is a subset of $S\times S\text{.}$

Equivalence relation, partition of a set $S\text{.}$ Equivalence class.

(Important Theorem) Let $S$ be a set.

(a)	Let $\sim$ be an equivalence relation on $S\text{.}$ The set of equivalence classes of the relation $\sim$ is a partition of $S\text{.}$
(b)	Let $\left\{S_{\alpha }\right\}$ be a partition of $S\text{.}$ The relation defined by $s\sim t$ if $s$ and $t$ are in the same $S_{\alpha }$ is an equivalence relation on $S\text{.}$

Orders

partially ordered set, totally ordered set, well ordered set.

upper/lower bound, $\text{sup}\left(E\right),$ $\text{inf}\left(E\right),$ $\text{min}\left(E\right),$ $\text{max}\left(E\right),$ maximal element, minimal element, smallest element, largest element.

Hasse diagram, lower/upper order ideal, intervals.

Ordered fields

An ordered field is a field $𝔽$ with a total order $\le$ such that

(a)	If $a,b,c\in 𝔽$ and $a\le b$ then $a+b\le b+c,$
(b)	If $a,b\in 𝔽$ and $a\ge 0$ and $b\ge 0$ then $ab\ge 0\text{.}$

Let $(𝔽,\le )$ be an ordered field.

(a)	If $a\in 𝔽$ and $a>0$ then $-a<0\text{.}$
(b)	If $a\in 𝔽$ and $a>0$ then $a^{-1}>0\text{.}$
(c)	If $a,b\in 𝔽$ and $a>0$ and $b>0$ then $ab>0\text{.}$
(d)	If $a\in 𝔽$ then $a^2$≥0.
(e)	If $a,b\in 𝔽$ and $a\ge 0$ and $b\ge 0$ then $a\le b$ if and only if $a^2\le b^2\text{.}$
(f)	$1\ge 0\text{.}$
(g)	If $a,b\in 𝔽$ and $a\ge 0$ and $b\ge 0$ then $a+b\ge 0\text{.}$

HW: Show that $ℝ$ with the usual order is an ordered field.

HW: Show that $ℂ$ is a field and there does not exist an order on $ℂ$ such that $ℂ$ is an ordered field.

Notes and References

These are a typed copy of Lecture 47 from a series of handwritten lecture notes for the class Metric and Hilbert Spaces given on October 20, 2014.

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