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 43: Kinds of spaces and Cauchy-Schwarz review

Topics

(1) Topological spaces, uniform spaces, metric spaces, normed vector spaces, inner product spaces.
(1.5) Examples of spaces: Subspaces and product spaces and B(V,W) and function spaces.
(2) Functions, Relations, Posets, Sets, functions, cardinality.
(3) Linear algebra – Vector spaces, bases, Linear transformations. Inner products, eigenvalues and eigenvectors.
(4) Convergence: Sequences and series, Hausdorff and compactness.
(5) Gram-Schmidt and determinants.

Modules of affine Lie algebras

(1) Category 𝒪
(2) Finite dimensional
(3) Wakimoto modules
(4) Extremal weight modules
(5) Smooth modules
(6) Admissible representations
(7) Weyl modules
Screaming operators are intertwiners

{Topological spaces} Mwith𝒯 | {Uniform spaces} Mwith𝔛 | {Metric spaces} Mwithd | {Normed vector spaces} Mwith· | {Inner vector spaces} Mwith·,·

Let 𝕂 be or and let A: be complex conjugation.

An inner product space is a vector space V over 𝕂 with a function V×V 𝕂 (x,y) x,y such that

(a) If x,y,zV then x+y,z=x,z+y,z,
(b) If x,yV then x,y=y,x,
(c) If c𝕂 and x,yV then cx,y=cx,y,
(d) If xV then x,x0,
(e) If xV and x,x=0 then x=0.

(5.1) Let V be an inner product space.

(a) If x,yV then x,yx·y.
(b) If x,yV then x+yx+y.

Let V be an inner product space. Define V 0 x x by x=x,x. Show that V with the function ·:V0 is a normed vector space.

Proof.

(a) Let x,yV.
Case 1: y=0. Then x,y=0 andx·y=0. Case 2: y0. Let a=x,x, b=x,y, c=y,y. Let λ𝕂. Then 0 x+λy,x+λy (by property (d)) = a+bλ+ bλ+cλ λ(by (a) and (b) and (c)). Let λ=-bc (using property (e)) so that 0a-bbc. Since c=y,y>0 (because y0), multiplying each side by c gives 0ac-b2= x2 y2- x,y2. So x,yx·y.
(b) By (a): Re(x,y) x,y xy so that x+y2 = x+y,x+y = x2+ y2+2 Re(x,y) x2+ y2+ 2x·y = (x+y)2. So x+yx+y.

HW: Prove the Cauchy-Schwartz and triangle inequalities for the norms p and q with p>0 and 1p+1q=1.

HW: Prove the Cauchy-Schwarz and triangle inequalities for the norms and 1.

HW: What is Lagrange's identity?

Notes and References

These are a typed copy of Lecture 43 from a series of handwritten lecture notes for the class Metric and Hilbert Spaces given on October 13, 2014.

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