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

$$\begin{array}{ccc}\left\{\text{Topological spaces}\right\}& & M\hspace{0.17em}\text{with}\hspace{0.17em}𝒯\\ \cup |\\ \left\{\text{Uniform spaces}\right\}& & M\hspace{0.17em}\text{with}\hspace{0.17em}𝔛\\ \cup |\\ \left\{\text{Metric spaces}\right\}& & M\hspace{0.17em}\text{with}\hspace{0.17em}d\\ \cup |\\ \left\{\text{Normed vector spaces}\right\}& & M\hspace{0.17em}\text{with}\hspace{0.17em}\Vert ·\Vert \\ \cup |\\ \left\{\text{Inner vector spaces}\right\}& & M\hspace{0.17em}\text{with}\hspace{0.17em}⟨·,·⟩\end{array}$$

Let $𝕂$ be $ℝ$ or $ℂ$ and let $\bar{\phantom{A}}:ℂ\to ℂ$ be complex conjugation.

An inner product space is a vector space $V$ over $𝕂$ with a function $$\begin{array}{ccc}V\times V& ⟶& 𝕂\\ (x,y)& ⟼& ⟨x,y⟩\end{array}$$ such that

(a)	If $x,y,z\in V$ then $⟨x+y,z⟩=⟨x,z⟩+⟨y,z⟩,$
(b)	If $x,y\in V$ then $⟨x,y⟩=\overline{⟨y,x⟩},$
(c)	If $c\in 𝕂$ and $x,y\in V$ then $⟨cx,y⟩=c⟨x,y⟩,$
(d)	If $x\in V$ then $⟨x,x⟩\in {ℝ}_{\ge 0}\text{,}$
(e)	If $x\in V$ and $⟨x,x⟩=0$ then $x=0\text{.}$

(5.1) Let $V$ be an inner product space.

(a)	If $x,y\in V$ then $\mid ⟨x,y⟩\mid \le \Vert x\Vert ·\Vert y\Vert \text{.}$
(b)	If $x,y\in V$ then $\Vert x+y\Vert \le \Vert x\Vert +\Vert y\Vert \text{.}$

Let $V$ be an inner product space. Define $$\begin{array}{ccc}V& ⟶& {ℝ}_{\ge 0}\\ x& ⟼& \Vert x\Vert \end{array}$$ by $$\Vert x\Vert =\sqrt{⟨x,x⟩}\text{.}$$ Show that $V$ with the function $\Vert ·\Vert :V\to {ℝ}_{\ge 0}$ is a normed vector space.

		Proof.
	(a) Let $x,y\in V\text{.}$ Case 1: $y=0\text{.}$ Then $$\mid ⟨x,y⟩\mid =0\text{and}\Vert x\Vert ·\Vert y\Vert =0\text{.}$$ Case 2: $y\ne 0\text{.}$ Let $$a=⟨x,x⟩,b=⟨x,y⟩,c=⟨y,y⟩\text{.}$$ Let $\lambda \in 𝕂\text{.}$ Then $$\begin{array}{rcl}{\displaystyle 0}& \le & {\displaystyle ⟨x+\lambda y,x+\lambda y⟩\text{(by property (d))}}\\ & =& {\displaystyle a+b\bar{\lambda }+\bar{b}\lambda +c\lambda \bar{\lambda }\text{(by (a) and (b) and (c)).}}\end{array}$$ Let $\lambda =-\frac{b}{c}$ (using property (e)) so that $$0\le a-\frac{b\bar{b}}{c}\text{.}$$ Since $c=⟨y,y⟩>0$ (because $y\ne 0\text{),}$ multiplying each side by $c$ gives $$0\le ac-{\mid b\mid }^2={\Vert x\Vert }^2{\Vert y\Vert }^2-{\mid ⟨x,y⟩\mid }^2\text{.}$$ So $\mid ⟨x,y⟩\mid \le \Vert x\Vert ·\Vert y\Vert \text{.}$ (b) By (a): $$\text{Re}\left(⟨x,y⟩\right)\le \mid ⟨x,y⟩\mid \le \Vert x\Vert \Vert y\Vert$$ so that $$\begin{array}{rcl}{\displaystyle {\Vert x+y\Vert }^2}& =& {\displaystyle ⟨x+y,x+y⟩}\\ & =& {\displaystyle {\Vert x\Vert }^2+{\Vert y\Vert }^2+2\text{Re}\left(⟨x,y⟩\right)}\\ & \le & {\displaystyle {\Vert x\Vert }^2+{\Vert y\Vert }^2+2\Vert x\Vert ·\Vert y\Vert }\\ & =& {\displaystyle {(\Vert x\Vert +\Vert y\Vert )}^2\text{.}}\end{array}$$ So $\Vert x+y\Vert \le \Vert x\Vert +\Vert y\Vert \text{.}$ $\square$

HW: Prove the Cauchy-Schwartz and triangle inequalities for the norms ${\Vert \hspace{0.17em}\Vert }_p$ and ${\Vert \hspace{0.17em}\Vert }_q$ with $p\in {ℝ}_{>0}$ and $\frac{1}{p}+\frac{1}{q}=1\text{.}$

HW: Prove the Cauchy-Schwarz and triangle inequalities for the norms ${\Vert \hspace{0.17em}\Vert }_{\infty }$ and ${\Vert \hspace{0.17em}\Vert }_1\text{.}$

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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