Lecture Notes in Functional Analysis

by William L. Paschke

edition 0.9

Contents

Frontmatter

I. Hilbert Space

II. Bounded Operators

III. Compact Operators

IV. The Spectral Theorem

Index and References

## D. Orthonormal Sets

Definition I.68

A subset $$$S$$$ of an inner product space $$$V$$$ is called orthonormal provided $$$\mathopen{}\left\lVert{}x\right\rVert\mathclose{}= 1$$$ and $$$x\perp y$$$ for all $$$x$$$ and $$$y$$$ in $$$S$$$.

Proposition I.69

Let $$$S= \mathopen{}\left\{\, {e}_{1}, \dotsc, {e}_{n}\,\right\}\mathclose{}$$$ be an orthonormal set in an inner product space $$$V$$$. For complex numbers $$${α}_{1}$$$, …, $$${α}_{n}$$$ and $$${β}_{1}$$$, …, $$${β}_{n}$$$, $$\mathopen{}\left\langle{}\sum_{i=1}^{n}{}{α}_{i}{e}_{i}, \sum_{j=1}^{n}{}{β}_{j}{e}_{j}\right\rangle\mathclose{}= \sum_{i=1}^{n}{}{α}_{i}\overline{{β}_{i}} \text{.}$$

Proof. Note that $$$\mathopen{}\left\langle{}{e}_{i}, {e}_{j}\right\rangle\mathclose{}= 0$$$ for $$$i\neq j$$$ and $$$\mathopen{}\left\langle{}{e}_{i}, {e}_{i}\right\rangle\mathclose{}= 1$$$.

Proposition I.70

Let $$$S= \mathopen{}\left\{\, {e}_{1}, \dotsc, {e}_{n}\,\right\}\mathclose{}$$$ be an orthonormal set in an inner product space $$$V$$$. Let $$$W= \operatorname{span}\mathopen{}\left( S\right)\mathclose{}$$$. Define a linear map $$$Q : V \to W$$$ by $$Q\mathopen{}\left( x\right)\mathclose{}= \sum_{i=1}^{n}{}\mathopen{}\left\langle{}x, {e}_{i}\right\rangle\mathclose{}{e}_{i} \text{.}$$ Then, for all $$$x\in V$$$, $$$x-Q\mathopen{}\left( x\right)\mathclose{}\in {W}^{\perp}$$$, $$$\mathopen{}\left\lVert{}Q\mathopen{}\left( x\right)\mathclose{}\right\rVert\mathclose{}\leq \mathopen{}\left\lVert{}x\right\rVert\mathclose{}$$$, $$$\mathopen{}\left\lVert{}x-Q\mathopen{}\left( x\right)\mathclose{}\right\rVert\mathclose{}\leq \mathopen{}\left\lVert{}x\right\rVert\mathclose{}$$$, and $$$Q\mathopen{}\left( x\right)\mathclose{}$$$ is the closest point in $$$W$$$ to $$$x$$$.

Proof. To prove the first claim, note that, for all $$$j$$$, $$$\mathopen{}\left\langle{}Q\mathopen{}\left( x\right)\mathclose{}, {e}_{j}\right\rangle\mathclose{}= \mathopen{}\left\langle{}x, {e}_{j}\right\rangle\mathclose{}$$$. Next, for the last claim, we can write $$$x= Q\mathopen{}\left( x\right)\mathclose{}+\mathopen{}\left(x-Q\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}$$$ with $$$Q\mathopen{}\left( x\right)\mathclose{}\in W$$$ and $$$x-Q\mathopen{}\left( x\right)\mathclose{}\in {W}^{\perp}$$$. Then $$${\mathopen{}\left\lVert{}x\right\rVert\mathclose{}}^{2}= {\mathopen{}\left\lVert{}Q\mathopen{}\left( x\right)\mathclose{}\right\rVert\mathclose{}}^{2}+{\mathopen{}\left\lVert{}x-Q\mathopen{}\left( x\right)\mathclose{}\right\rVert\mathclose{}}^{2}$$$. Finally, for any $$$w\in W$$$, $$$\mathopen{}\left\lVert{}x-w\right\rVert\mathclose{}\geq \mathopen{}\left\lVert{}\mathopen{}\left(x-w\right)\mathclose{}-\mathopen{}\left(Q\mathopen{}\left( x-w\right)\mathclose{}\right)\mathclose{}\right\rVert\mathclose{}= \mathopen{}\left\lVert{}x-w-Q\mathopen{}\left( x\right)\mathclose{}+Q\mathopen{}\left( w\right)\mathclose{}\right\rVert\mathclose{}$$$. Since $$$w\in W$$$, $$$Q\mathopen{}\left( w\right)\mathclose{}= w$$$.

Proposition I.71 (Gram-Schmidt Procedure)

Let $$${x}_{1}$$$, $$${x}_{2}$$$, … be a linearly independent sequence in an inner product space $$$V$$$. Then there exists an orthonormal sequence $$${e}_{1}$$$, $$${e}_{2}$$$, …, in $$$V$$$ such that $$$\operatorname{span}\mathopen{}\left( \mathopen{}\left\{\, {x}_{1}, {x}_{2}, \dotsc\,\right\}\mathclose{}\right)\mathclose{}= \operatorname{span}\mathopen{}\left( \mathopen{}\left\{\, {e}_{1}, {e}_{2}, \dotsc\,\right\}\mathclose{}\right)\mathclose{}$$$.

Proof. Start with $$${e}_{1}= \frac{{x}_{1}}{\mathopen{}\left\lVert{}{x}_{1}\right\rVert\mathclose{}}$$$. Inductive step: Let $${y}_{n+1}= {x}_{n+1}-\sum_{i=1}^{n}{}\mathopen{}\left\langle{}{x}_{n+1}, {e}_{i}\right\rangle\mathclose{}{e}_{i}$$ (which is not zero by the inductive hypothesis). Then let $$${e}_{n+1}= \frac{{y}_{n+1}}{\mathopen{}\left\lVert{}{y}_{n+1}\right\rVert\mathclose{}}$$$.

Proposition I.72 (Bessel's Inequality)

Let $$${e}_{1}$$$, $$${e}_{2}$$$, … be an orthonormal sequence in an inner product space $$$V$$$. Then $$\sum_{i=1}^{\infty}{}{\mathopen{}\left\lvert{}\mathopen{}\left\langle{}x, {e}_{i}\right\rangle\mathclose{}\right\rvert\mathclose{}}^{2}\leq {\mathopen{}\left\lVert{}x\right\rVert\mathclose{}}^{2}$$ for all $$$x$$$ in $$$V$$$. If $$$V$$$ is complete, then $$$\sum_{i=1}^{\infty}{}\mathopen{}\left\langle{}x, {e}_{i}\right\rangle\mathclose{}{e}_{i}$$$ converges and sums to the orthogonal projection of $$$x$$$ on $$$\overline{\operatorname{span}\mathopen{}\left( \mathopen{}\left\{\, {e}_{1}, {e}_{2}, \dotsc\,\right\}\mathclose{}\right)\mathclose{}}$$$.

Proof. The first claim follows from Proposition I.70 since $$\sum_{n=1}^{N}{}{\mathopen{}\left\lvert{}\mathopen{}\left\langle{}x, {e}_{n}\right\rangle\mathclose{}\right\rvert\mathclose{}}^{2}\leq {\mathopen{}\left\lVert{}x\right\rVert\mathclose{}}^{2}$$ for all $$$N$$$.

For the second claim, assume $$$V= H$$$ is a Hilbert space. For $$$M\gt N$$$, we have $${\mathopen{}\left\lVert{}\sum_{n=N}^{M}{}\mathopen{}\left\langle{}x, {e}_{n}\right\rangle\mathclose{}{e}_{n}\right\rVert\mathclose{}}^{2}= \sum_{n=N}^{M}{}{\mathopen{}\left\lvert{}\mathopen{}\left\langle{}x, {e}_{n}\right\rangle\mathclose{}\right\rvert\mathclose{}}^{2} \text{,}$$ which is small for large $$$N$$$ and $$$M$$$ by the first part. So, the sequence of partial sums for the whole series, $$$\sum_{n=1}^{\infty}{}\mathopen{}\left\langle{}x, {e}_{n}\right\rangle\mathclose{}{e}_{n}$$$, is Cauchy, and so the series converges, say to $$$w$$$ in $$$H$$$. Then $$$w= \sum_{n=1}^{\infty}{}\mathopen{}\left\langle{}x, {e}_{n}\right\rangle\mathclose{}{e}_{n}$$$ and $$\mathopen{}\left\langle{}w, {e}_{j}\right\rangle\mathclose{}= \sum_{n=1}^{\infty}{} \mathopen{}\left\langle{}x, {e}_{n}\right\rangle\mathclose{}\mathopen{}\left\langle{}{e}_{n}, {e}_{j}\right\rangle\mathclose{}$$ (continuity of $$$\mathopen{}\left\langle{}\cdot, {e}_{j}\right\rangle\mathclose{}$$$ comes from Proposition I.3). So $$$x-w\perp {e}_{j}$$$ and $$$x-w\perp \operatorname{span}\mathopen{}\left( \mathopen{}\left\{\, {e}_{1}, {e}_{2}, \dotsc\,\right\}\mathclose{}\right)\mathclose{}$$$. Further, $$$w$$$ is in $$$\overline{\operatorname{span}\mathopen{}\left( \mathopen{}\left\{\, {e}_{1}, {e}_{2}, \dotsc\,\right\}\mathclose{}\right)\mathclose{}}$$$.

Theorem I.73

Let $$$H$$$ be a separable infinite-dimensional Hilbert space. Then there exists an orthonormal set $$$\mathopen{}\left\{\, {e}_{1}, {e}_{2}, \dotsc\,\right\}\mathclose{}$$$ with $$$\overline{\operatorname{span}\mathopen{}\left( \mathopen{}\left\{\, {e}_{1}, {e}_{2}, \dotsc\,\right\}\mathclose{}\right)\mathclose{}}= H$$$. Further, $$x= \sum_{i=1}^{\infty}{}\mathopen{}\left\langle{}x, {e}_{i}\right\rangle\mathclose{}{e}_{i}$$ for all $$$x\in H$$$ and $$\mathopen{}\left\langle{}x, y\right\rangle\mathclose{}= \sum_{i=1}^{\infty}{}\mathopen{}\left\langle{}x, {e}_{i}\right\rangle\mathclose{}\mathopen{}\left\langle{}{e}_{i}, y\right\rangle\mathclose{}$$ for all $$$x\in H$$$ and $$$y\in H$$$.

Proof. From the hypothesis and the size of $$$H$$$, we get a dense set $$$\mathopen{}\left\{\, {x}_{1}, {x}_{2}, \dotsc\,\right\}\mathclose{}$$$ in $$$H$$$ and from it a linearly independent sequence $$$\mathopen{}\left\{\, {w}_{1}, {w}_{2}, \dotsc\,\right\}\mathclose{}$$$ whose linear span is dense in $$$H$$$. Apply Proposition I.71 to get an orthonormal sequence $$$\mathopen{}\left\{\, {e}_{1}, {e}_{2}, \dotsc\,\right\}\mathclose{}$$$ with the same closed span, namely $$$H$$$.

The first claim follows from Proposition I.72. For the second claim, let $$${x}_{n}= \sum_{i=1}^{n}{}\mathopen{}\left\langle{}x, {e}_{i}\right\rangle\mathclose{}{e}_{i}$$$ and $$${y}_{n}= \sum_{j=1}^{\infty}{}\mathopen{}\left\langle{}y, {e}_{j}\right\rangle\mathclose{}\mathopen{}\left\langle{}{e}_{j}, y\right\rangle\mathclose{}$$$. So, $$$\mathopen{}\left\langle{}{x}_{n}, {y}_{n}\right\rangle\mathclose{}= \sum_{i=1}^{n}{}\mathopen{}\left\langle{}x, {e}_{i}\right\rangle\mathclose{}\mathopen{}\left\langle{}{e}_{i}, y\right\rangle\mathclose{}$$$ by Proposition I.70. Let $$$n$$$ go to $$$\infty$$$ and notice $$$\mathopen{}\left\langle{}{x}_{n}, {y}_{n}\right\rangle\mathclose{}$$$ converges to $$$\mathopen{}\left\langle{}x, y\right\rangle\mathclose{}$$$ by the continuity of the inner product, which comes from the estimate $$\mathopen{}\left\lvert{}\mathopen{}\left\langle{}{x}_{n}, {y}_{n}\right\rangle\mathclose{}-\mathopen{}\left\langle{}x, y\right\rangle\mathclose{}\right\rvert\mathclose{}\leq \mathopen{}\left\lvert{}\mathopen{}\left\langle{}{x}_{n}-x, {y}_{n}\right\rangle\mathclose{}\right\rvert\mathclose{}+\mathopen{}\left\lvert{}\mathopen{}\left\langle{}x, {y}_{n}-y\right\rangle\mathclose{}\right\rvert\mathclose{}\leq \mathopen{}\left\lVert{}{x}_{n}-x\right\rVert\mathclose{}\mathopen{}\left\lVert{}{y}_{n}\right\rVert\mathclose{}+\mathopen{}\left\lVert{}x\right\rVert\mathclose{}\mathopen{}\left\lVert{}{y}_{n}-y\right\rVert\mathclose{}\text{.}$$

Definition I.74

Let $$$H$$$ be a separable infinite-dimensional Hilbert space. An orthonormal set $$$\mathopen{}\left\{\, {e}_{1}, {e}_{2}, \dotsc\,\right\}\mathclose{}$$$ such that $$$\overline{\operatorname{span}\mathopen{}\left( \mathopen{}\left\{\, {e}_{1}, {e}_{2}, \dotsc\,\right\}\mathclose{}\right)\mathclose{}}= H$$$ is an orthonormal basis of $$$H$$$.

Definition I.75

An isomorphism of Hilbert spaces $$$H\simeq K$$$ is an inner-product-preserving linear surjection.

Theorem I.76

If $$$H$$$ is a separable, infinite dimensional Hilbert space with orthonormal basis $$$\mathopen{}\left\{\, {e}_{1}, {e}_{2}, \dotsc\,\right\}\mathclose{}$$$, then $$$U : H \to \mathrm{l}^{0}$$$ defined by $$$U\mathopen{}\left( x\right)\mathclose{}= \mathopen{}\left(\mathopen{}\left\langle{}x, {e}_{n}\right\rangle\mathclose{}\right)\mathclose{}_{n=1}^{\infty}$$$ is an isomorphism.

Proof. Notice that $$$\mathopen{}\left\langle{}U\mathopen{}\left( x\right)\mathclose{}, U\mathopen{}\left( y\right)\mathclose{}\right\rangle\mathclose{}= \mathopen{}\left\langle{}x, y\right\rangle\mathclose{}$$$ by Theorem I.73. Furthermore, $$$U$$$ is onto because $$$\mathopen{}\left({α}_{n}\right)\mathclose{}_{n=1}^{\infty}\in \mathrm{l}^{0}$$$ makes $$$\sum_{n=1}^{\infty}{}{α}_{n}{e}_{n}$$$ converge in $$$H$$$. Since $${\mathopen{}\left\lVert{}\sum_{n=N}^{M}{}{α}_{n}{e}_{n}\right\rVert\mathclose{}}^{2}= \sum_{n=N}^{M}{}{\mathopen{}\left\lvert{}{α}_{n}\right\rvert\mathclose{}}^{2} \text{,}$$ the sequence of partial sums is Cauchy. Thus we have $$$U\mathopen{}\left( \sum_{n=1}^{\infty}{}{α}_{n}{e}_{n}\right)\mathclose{}= \mathopen{}\left({α}_{n}\right)\mathclose{}_{n=1}^{\infty}$$$.

Example I.77

Let's look at the sine series in $$$\mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{}$$$. Consider $$${e}_{n}\mathopen{}\left( t\right)\mathclose{}= \sqrt{2}\sin\mathopen{}\left( n\mathrm{\pi}t\right)\mathclose{}$$$ for $$$n\in \mathbb{N}$$$. From calculus its easy to see that $$$\mathopen{}\left\{\, {e}_{1}, {e}_{2}, \dotsc\,\right\}\mathclose{}$$$ is orthonormal. We claim that it is in fact dense in $$$\mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{}$$$. Take $$$f\in \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{}$$$ such that $$$\mathopen{}\left\langle{}f, {e}_{n}\right\rangle\mathclose{}= 0$$$ for all $$$n$$$. Extend $$$f$$$ to $$$\mathopen{}\left[{-}1, 1\right]\mathclose{}$$$ by $$$f\mathopen{}\left( {-}t\right)\mathclose{}= {-}f\mathopen{}\left( t\right)\mathclose{}$$$ for $$$0\lt t\lt 1$$$. Consider also the function $$$g$$$ defined (almost everywhere) on the unit circle $$$\mathbb{T}= \mathopen{}\left\{\, z\in \mathbb{C}\,\middle\vert\, \mathopen{}\left\lvert{}z\right\rvert\mathclose{}= 1\,\right\}\mathclose{}$$$ by $$$g\mathopen{}\left( {\mathrm{e}}^{\mathrm{\pi}\mathrm{i}θ}\right)\mathclose{}= f\mathopen{}\left( θ\right)\mathclose{}$$$. Notice $$\int _{\mathbb{T}}{}g\mathopen{}\left( z\right)\mathclose{}{z}^{n}\,\mathrm{d}z= \int _{{-}1}^{1}{}f\mathopen{}\left( θ\right)\mathclose{}{\mathrm{e}}^{\mathrm{\pi}\mathrm{i}nθ}\,\mathrm{d}θ= \int _{{-}1}^{1}{} f\mathopen{}\left( θ\right)\mathclose{}\mathopen{}\left(\cos\mathopen{}\left( n\mathrm{\pi}θ\right)\mathclose{}+\mathrm{i}\sin\mathopen{}\left( n\mathrm{\pi}θ\right)\mathclose{}\right)\mathclose{} \,\mathrm{d}θ= 0$$ because $$$f$$$ is an odd function and $$$\mathopen{}\left\langle{}f, {e}_{n}\right\rangle\mathclose{}= 0$$$ for all $$$n$$$.

So we have that $$$\int _{\mathbb{T}}{}g\mathopen{}\left( z\right)\mathclose{}φ\mathopen{}\left( z\right)\mathclose{}\,\mathrm{d}z= 0$$$ for all $$$φ\in \mathrm{C}\mathopen{}\left( \mathbb{T}\right)\mathclose{}$$$ because polynomials in $$$z$$$ and $$${z}^{-1}$$$ are uniformly dense in $$$\mathrm{C}\mathopen{}\left( \mathbb{T}\right)\mathclose{}$$$ by the Stone-Weierstrass Theorem and $$$g= g\mathbf{1}$$$ in $$$\mathrm{L}^{\mathrm{1}}\mathopen{}\left( \mathbb{T}\right)\mathclose{}$$$. If a sequence $$$\mathopen{}\left({φ}_{n}\right)\mathclose{}_{n=1}^{\infty}$$$ converges to $$$φ$$$ uniformly, then $$\mathopen{}\left\lvert{}\int _{\mathbb{T}}{}g\mathopen{}\left( z\right)\mathclose{}{φ}_{n}\mathopen{}\left( z\right)\mathclose{}\,\mathrm{d}z-\int _{\mathbb{T}}{}g\mathopen{}\left( z\right)\mathclose{}φ\mathopen{}\left( z\right)\mathclose{}\,\mathrm{d}z\right\rvert\mathclose{}\leq g\mathopen{}\left\lVert{}{φ}_{n}-φ\right\rVert_\infty\mathclose{} \text{.}$$

For any arc $$$A$$$ in $$$\mathbb{T}$$$, $$$\chi_{A}$$$ is the $$$\mathrm{L}^{\mathrm{2}}$$$-limit of a sequence of continuous functions on $$$\mathbb{T}$$$, so $$$\int _{\mathbb{T}}{}g\mathopen{}\left( z\right)\mathclose{}\chi_{A}\mathopen{}\left( z\right)\mathclose{}\,\mathrm{d}z= 0$$$. For every measurable subset $$$S\subseteq \mathbb{T}$$$ we get a sequence $$$\mathopen{}\left({U}_{n}\right)\mathclose{}_{n=1}^{\infty}$$$ of countable unions of open arcs with $$$A$$$ and the measure of $$${U}_{n}\setminus S$$$ converges to zero by the regularity of Lebesque measure. So $$$\int _{{U}_{n}}{}g\mathopen{}\left( z\right)\mathclose{}\,\mathrm{d}z= 0$$$ for all $$$n$$$ implies $$$\int _{S}{}g\mathopen{}\left( z\right)\mathclose{}\,\mathrm{d}z= 0$$$. Thus if $$$g$$$ is zero almost everywhere, then $$$f$$$ is zero almost everywhere.

We have shown that $$$f= \sum_{n=1}^{\infty}{}\mathopen{}\left\langle{}f, {e}_{n}\right\rangle\mathclose{}{e}_{n}$$$, and thus $$${\mathopen{}\left\lVert{}f\right\rVert\mathclose{}}^{2}= \sum_{n=1}^{\infty}{}{\mathopen{}\left\lvert{}\mathopen{}\left\langle{}f, {e}_{n}\right\rangle\mathclose{}\right\rvert\mathclose{}}^{2}$$$ for all $$$f\in \mathrm{L}^{\mathrm{2}}$$$. For any choice of $$$f$$$ that is readily integrable against each $$${e}_{n}$$$, we get an amusing series summation. For instance, $$$f\equiv \mathbf{1}$$$ gives $$\mathopen{}\left\langle{}f, {e}_{n}\right\rangle\mathclose{}= \sqrt{2}\int _{0}^{1}{}\sin\mathopen{}\left( n\mathrm{\pi}t\right)\mathclose{}\,\mathrm{d}t= \begin{cases}\frac{2\sqrt{2}}{\mathrm{\pi}n}, & n|\text{odd}; \\ 0, & n|\text{even.}\end{cases}$$ Then $$${\mathopen{}\left\lVert{}f\right\rVert\mathclose{}}^{2}= \sum_{n=1}^{\infty}{}{\mathopen{}\left\lvert{}\mathopen{}\left\langle{}f, {e}_{n}\right\rangle\mathclose{}\right\rvert\mathclose{}}^{2}$$$, so $$1= \frac{8}{{\mathrm{\pi}}^{2}}\sum_{n|\text{odd}}{}\frac{1}{{n}^{2}}$$ implies $$\sum_{n|\text{odd}}{}\frac{1}{{n}^{2}} = \frac{{\mathrm{\pi}}^{2}}{8}$$and $$\sum_{n=1}^{\infty}{}\frac{1}{{n}^{2}} = \frac{4}{3}\mathopen{}\left(\frac{{\mathrm{\pi}}^{2}}{8}\right)\mathclose{}= \frac{{\mathrm{\pi}}^{2}}{6} \text{.}$$