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 33: Inner product spaces and orthogonality

An inner product space is a vector space $V$ over $ℂ$ with a function $$\begin{array}{ccc}V\times V& ⟶& ℂ\\ (v_1,v_2)& ⟼& ⟨v_1,v_2⟩\end{array}$$ such that

(a)	If $v_1,v_2,v_3\in V$ and $c_1,c_2\in ℂ$ then $$⟨c_1{v}_1+c_2{v}_2,v_3⟩=c_1⟨v_1,v_3⟩+c_2⟨v_2,v_3⟩\text{.}$$
(b)	If $v_1,v_2,v_3\in V$ and $c_1,c_2\in ℂ$ then $$⟨v_3,c_1{v}_1+c_2{v}_2⟩=\overline{c_1}⟨v_3,v_1⟩+\overline{c_2}⟨v_3,v_2⟩\text{.}$$
(c)	If $v_1,v_2\in V$ then $⟨v_2,v_1⟩=⟨\overline{v_1,v_2}⟩\text{.}$
(d)	If $v\in V$ and $⟨v,v⟩=0$ then $v=0\text{.}$
(e)	If $v\in V$ then $⟨v,v⟩\in {ℝ}_{\ge 0}\text{.}$

Let $(V,⟨\hspace{0.17em}⟩)$ be an inner product space. Define $\Vert \hspace{0.17em}\Vert :V\to {ℝ}_{\ge 0}$ by $$\Vert v\Vert =\sqrt{⟨v,v⟩}\text{.}$$

HW: Show that $(V,\Vert ·\Vert )$ is a normed vector space.

A Hilbert space is an inner product space $(V,⟨\hspace{0.17em}⟩)$ such that $V$ is a complete metric space.

Orthogonal complements

Let $(V,⟨\hspace{0.17em}⟩)$ be an inner product space and let $W\subseteq V$ be a subspace of $V\text{.}$

The orthogonal complement of $W$ in $V$ is $$W^{\perp }=\{v\in V\hspace{0.17em}|\hspace{0.17em}\text{if}\hspace{0.17em}w\in W\hspace{0.17em}\text{then}\hspace{0.17em}⟨v,w⟩=0\}\text{.}$$

If $V$ is a Hilbert space and $W$ is closed then $$V=W\oplus W^{\perp }\text{.}$$

Let $V$ be an inner product space and let $W$ be a subspace of $V\text{.}$ An orthogonal projection onto $W$ is a linear transformation $P:V\to V$ such that

(a)	if $v\in V$ then $P\left(v\right)\in W,$
(b)	if $v\in W$ then $v-P\left(v\right)\in W^{\perp }\text{.}$

(sub-Theorem) Let $V$ be a Hilbert space and let $W$ be a subspace of $V\text{.}$ There exists an orthogonal projection $P:V\to V$ onto $W$ if and only if $W$ is closed.

Orthogonality

HW: Let $(V,⟨\hspace{0.17em}⟩)$ be an inner product space. Show that $$\begin{array}{cccc}d:& V& ⟶& V^{*}\\ & w& ⟼& {\phi }_w:& V& ⟶& ℂ\\ & & & & v& ⟼& ⟨v,w⟩\end{array}$$ is a linear transformation. Let $W$ be a subspace of $V\text{.}$ Show that $$W^{\perp }=\bigcap _{w\in W}\text{ker}\left({\phi }_w\right)\text{.}$$

Let $V$ be a Hilbert space.

An orthonormal sequence in $V$ is a sequence $a_1,a_2,a_3,\dots$ in $V$ such that if $i,j\in {\mathbb{Z}}_{>0}$ then $$⟨a_i,a_j⟩=\{\begin{array}{ll}0,& \text{if}\hspace{0.17em}i\ne j,\\ 1,& \text{if}\hspace{0.17em}i=j\text{.}\end{array}$$

HW: Let $a_1,a_2,\dots$ be an orthonormal sequence in $V\text{.}$ Let $W=\text{span}\{a_1,a_2,\dots \}\text{.}$ Show that

(a)	If $v\in V$ then $$\sum _{n\in {\mathbb{Z}}_{\ge 0}}{\mid ⟨v,a_n⟩\mid }^2\le {\Vert v\Vert }^2\text{.}\text{(Bessel's inequality)}$$
(b)	$P:V\to V$ given by $$P\left(v\right)=\sum _{n\in {\mathbb{Z}}_{>0}}⟨v,a_n⟩a_n$$ is an orthogonal projection onto $\bar{W}\text{.}$
(c)	If $\bar{W}=V$ then $\{a_1,a_2,a_3,\dots \}$ is a Schauder basis of $V$ i.e., every $v\in V$ can be written uniquely as $v={\lambda }_1{a}_1+{\lambda }_2{a}_2+\cdots \text{.}$

HW: (Fourier analysis) Let $e_0,e_1,e_{-1},e_2,e_{-2},\dots$ in $L^2[0,2\pi ]$ be given by $$e_m\left(t\right)=\frac{1}{\sqrt{2\pi }}{e}^{\text{im}\hspace{0.17em}t}\text{.}$$ Show that $e_0,e_1,e_{-1},e_2,e_{-2},\dots$ is an orthonormal basis of $L^2[0,2\pi ]\text{.}$

Gram-Schmidt

Let $v_1,v_2,\dots$ be a sequence of linearly independent vectors in $V\text{.}$ Define $$a_1=\frac{v_1}{\Vert v_1\Vert }\text{and}{a}_{n+1}=\frac{v_{n+1}-⟨v_{n+1},a_1⟩a_1-\cdots -⟨v_{n+1},a_n⟩a_n}{\Vert v_{n+1}-⟨v_{n+1},a_1⟩a_1-\cdots -⟨v_{n+1},a_n⟩a_n\Vert }\text{.}$$ Then $a_1,a_2,\dots$ is an orthonormal sequence in $V\text{.}$

Notes and References

These are a typed copy of Lecture 33 from a series of handwritten lecture notes for the class Metric and Hilbert Spaces given on September 23, 2014.

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