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, .

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:ST 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 Card()=Card()=Card(>0)Card().

HW: Show that Card()=Card(2).

Relations

Let S be a set. A relation on S is a subset of S×S.

Equivalence relation, partition of a set S. Equivalence class.

(Important Theorem) Let S be a set.

(a) Let be an equivalence relation on S. The set of equivalence classes of the relation is a partition of S.
(b) Let {Sα} be a partition of S. The relation defined by st if s and t are in the same Sα is an equivalence relation on S.

Orders

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

upper/lower bound, sup(E), inf(E), min(E), max(E), 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 such that

(a) If a,b,c𝔽 and ab then a+bb+c,
(b) If a,b𝔽 and a0 and b0 then ab0.

Let (𝔽,) be an ordered field.

(a) If a𝔽 and a>0 then -a<0.
(b) If a𝔽 and a>0 then a-1>0.
(c) If a,b𝔽 and a>0 and b>0 then ab>0.
(d) If a𝔽 then a20.
(e) If a,b𝔽 and a0 and b0 then ab if and only if a2b2.
(f) 10.
(g) If a,b𝔽 and a0 and b0 then a+b0.

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

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