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

## I. Hilbert Space

Definition I.1

Let $$$V$$$ be a vector space over the complex numbers $$$\mathbb{C}$$$. A positive sesquilinear form on $$$V$$$ is a map $$$\mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{} : V\times V \to \mathbb{C}$$$ (so that $$$\mathopen{}\left(x, y\right)\mathclose{}\mapsto \mathopen{}\left\langle{}x, y\right\rangle\mathclose{}$$$) that satisfies the following conditions for all $$$w$$$, $$$x$$$, and $$$y$$$ in $$$V$$$ and $$$λ$$$ in $$$\mathbb{C}$$$:

1. $$$\mathopen{}\left\langle{}x+λy, w\right\rangle\mathclose{}= \mathopen{}\left\langle{}x, w\right\rangle\mathclose{}+λ\mathopen{}\left\langle{}y, w\right\rangle\mathclose{}$$$.
2. $$$\overline{ \mathopen{}\left\langle{}x, y\right\rangle\mathclose{} }= \mathopen{}\left\langle{}y, x\right\rangle\mathclose{}$$$.
3. $$$\mathopen{}\left\langle{}x, x\right\rangle\mathclose{}\geq 0$$$.

Remark I.2

Satisfying the first property in Definition I.1 is often refered to as being linear in the first slot. The first two properties together imply that a positive sesquilinear form is also conjugate linear in the second slot, meaning $$$\mathopen{}\left\langle{}w, x+λy\right\rangle\mathclose{}= \mathopen{}\left\langle{}w, x\right\rangle\mathclose{}+\overline{ λ }\mathopen{}\left\langle{}w, y\right\rangle\mathclose{}$$$.

Proposition I.3 (Cauchy-Schwarz Inequality)

Let $$$V$$$ be a vector space with positive sesquilinear form $$$\mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{}$$$. For all $$$x$$$ and $$$y$$$ in $$$V$$$, $$${\mathopen{}\left\lvert{}\mathopen{}\left\langle{}x, y\right\rangle\mathclose{}\right\rvert\mathclose{}}^{2}\leq \mathopen{}\left\langle{}x, x\right\rangle\mathclose{}\mathopen{}\left\langle{}y, y\right\rangle\mathclose{}$$$.

Proof. For all $$$x$$$ and $$$y$$$ in $$$V$$$, $$$0\leq \mathopen{}\left\langle{}x+λy, x+λy\right\rangle\mathclose{}= \mathopen{}\left\langle{}x, x\right\rangle\mathclose{}+λ\mathopen{}\left\langle{}y, x\right\rangle\mathclose{}+\overline{λ}\mathopen{}\left\langle{}x, y\right\rangle\mathclose{}+{\mathopen{}\left\lvert{}λ\right\rvert\mathclose{}}^{2}\mathopen{}\left\langle{}y, y\right\rangle\mathclose{}$$$.

If $$$\mathopen{}\left\langle{}y, y\right\rangle\mathclose{}= 0$$$, then $$$0\leq \mathopen{}\left\langle{}x, x\right\rangle\mathclose{}+2\Re\mathopen{}\left( \overline{λ}\mathopen{}\left\langle{}x, y\right\rangle\mathclose{}\right)\mathclose{}$$$ for all complex numbers $$$λ$$$, and, in particular, for $$$λ= {-} t \mathopen{}\left\langle{}x, y\right\rangle\mathclose{}$$$, which implies that $$$\mathopen{}\left\langle{}x, y\right\rangle\mathclose{}= 0$$$.

If $$$\mathopen{}\left\langle{}y, y\right\rangle\mathclose{}\neq 0$$$, let $$$λ= {-} \frac{\mathopen{}\left\langle{}x, y\right\rangle\mathclose{}}{\mathopen{}\left\langle{}y, y\right\rangle\mathclose{}}$$$. Then $$0\leq \mathopen{}\left\langle{}x, x\right\rangle\mathclose{}-2\frac{{\mathopen{}\left\lvert{}\mathopen{}\left\langle{}x, y\right\rangle\mathclose{}\right\rvert\mathclose{}}^{2}}{\mathopen{}\left\langle{}y, y\right\rangle\mathclose{}}+\frac{{\mathopen{}\left\lvert{}\mathopen{}\left\langle{}x, y\right\rangle\mathclose{}\right\rvert\mathclose{}}^{2}}{\mathopen{}\left\langle{}y, y\right\rangle\mathclose{}}= \mathopen{}\left\langle{}x, x\right\rangle\mathclose{}-\frac{{\mathopen{}\left\lvert{}\mathopen{}\left\langle{}x, y\right\rangle\mathclose{}\right\rvert\mathclose{}}^{2}}{\mathopen{}\left\langle{}y, y\right\rangle\mathclose{}} \text{.}$$

Proposition I.4 (Triangle Inequality)

Let $$$V$$$ be a vector space with positive sesquilinear form $$$\mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{}$$$. For all $$$x$$$ and $$$y$$$ in $$$V$$$, $$${\mathopen{}\left\langle{}x+y, x+y\right\rangle\mathclose{}}^{\frac{1}{2}}\leq {\mathopen{}\left\langle{}x, x\right\rangle\mathclose{}}^{\frac{1}{2}}+{\mathopen{}\left\langle{}y, y\right\rangle\mathclose{}}^{\frac{1}{2}}$$$.

Proof. Using Proposition I.3, $$\mathopen{}\left\langle{}x+y, x+y\right\rangle\mathclose{}= \mathopen{}\left\langle{}x, x\right\rangle\mathclose{}+2\Re\mathopen{}\left( \mathopen{}\left\langle{}x, y\right\rangle\mathclose{}\right)\mathclose{}+\mathopen{}\left\langle{}y, y\right\rangle\mathclose{}\leq \mathopen{}\left\langle{}x, x\right\rangle\mathclose{}+2{\mathopen{}\left\langle{}x, x\right\rangle\mathclose{}}^{\frac{1}{2}}{\mathopen{}\left\langle{}y, y\right\rangle\mathclose{}}^{\frac{1}{2}}+\mathopen{}\left\langle{}y, y\right\rangle\mathclose{}= {\mathopen{}\left({\mathopen{}\left\langle{}x, x\right\rangle\mathclose{}}^{\frac{1}{2}}+{\mathopen{}\left\langle{}y, y\right\rangle\mathclose{}}^{\frac{1}{2}}\right)\mathclose{}}^{2} \text{.}$$

Definition I.5

An inner product $$$\mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{}$$$ on a vector space $$$V$$$ is a positive sesquilinear form such that $$$\mathopen{}\left\langle{}x, x\right\rangle\mathclose{}= 0$$$ if and only if $$$x= 0$$$. A vector space equipped with an inner product is an inner product space.

Example I.6

Let $$$V$$$ equal $$${\mathbb{C}}^{n}$$$. Given an $$$n$$$-by-$$$n$$$ matrix $$$M$$$ that is positive definite (meaning $$$M$$$ is invertible and $$$Mu\cdot u\geq 0$$$ for all $$$u$$$ in $$$V$$$), define $$$\mathopen{}\left\langle{}u, v\right\rangle\mathclose{}= Mu\cdot v$$$ where $$$u\cdot v$$$ is the usual dot product of vectors $$$u$$$ and $$$v$$$ in $$${\mathbb{C}}^{n}$$$.

Example I.7

Let $$$V$$$ be $$$\mathrm{C}\mathopen{}\left( \mathopen{}\left[a, b\right]\mathclose{}, \mathbb{C}\right)\mathclose{}$$$, the set of continuous complex-valued functions on the real interval $$$\mathopen{}\left[a, b\right]\mathclose{}$$$. For functions $$$φ$$$ and $$$ψ$$$ in $$$V$$$, define $$$\mathopen{}\left\langle{}φ, ψ\right\rangle\mathclose{}$$$ to be $$$\int _{ a}^{ b}{} φ\mathopen{}\left( t\right)\mathclose{}\overline{ ψ\mathopen{}\left( t\right)\mathclose{} } \,\mathrm{d}t$$$.

Example I.8

Let $$$V$$$ be the set of complex sequences that are eventually zero. For two such sequences $$$\mathopen{}\left({α}_{n}\right)\mathclose{}_{n=1}^{\infty}$$$ and $$$\mathopen{}\left({β}_{n}\right)\mathclose{}_{n=1}^{\infty}$$$, define $$\mathopen{}\left\langle{}\mathopen{}\left({α}_{n}\right)\mathclose{}_{n=1}^{\infty}, \mathopen{}\left({β}_{n}\right)\mathclose{}_{n=1}^{\infty}\right\rangle\mathclose{}= \sum_{j=1}^{\infty}{} {α}_{j}\overline{{β}_{j}} \text{.}$$

Example I.9

Riffing on the previous examples, take $$$V$$$ as in Example I.8 but define the inner product by $$\mathopen{}\left\langle{}\mathopen{}\left({α}_{n}\right)\mathclose{}_{n=1}^{\infty}, \mathopen{}\left({β}_{n}\right)\mathclose{}_{n=1}^{\infty}\right\rangle\mathclose{}= \int _{0}^{1}{} \mathopen{}\left(\sum_{n=1}^{\infty}{} {α}_{n}{t}^{n} \right)\mathclose{}\overline{ \mathopen{}\left(\sum_{m=1}^{\infty}{} {β}_{m}{t}^{m} \right)\mathclose{} } \,\mathrm{d}t= \sum_{m,n=1}^{\infty}{} \frac{{α}_{n}\overline{ {α}_{m} }}{m+n+1} \text{.}$$

For subsets $$$R$$$ and $$$S$$$ of a complex vector space $$$V$$$, we will use the notation $$\mathbb{C}S= \mathopen{}\left\{\, λy \,\middle\vert\, , λ\in \mathbb{C}, , y\in S, \,\right\}\mathclose{}$$ for the set of all scalar multiples of elements of $$$S$$$ and $$R+S= \mathopen{}\left\{\, x+y \,\middle\vert\, , x\in R, , y\in S, \,\right\}\mathclose{}$$ for the set of all possible sums of elements from $$$R$$$ and $$$S$$$. If $$$V$$$ has a positive sesquilinear form or is an inner product space, then $$\mathopen{}\left\langle{}R, S\right\rangle\mathclose{}= \mathopen{}\left\{\, \mathopen{}\left\langle{}x, y\right\rangle\mathclose{} \,\middle\vert\, , x\in R, , y\in S, \,\right\}\mathclose{}$$ is the set of all possible inner products.

Definition I.10

A vector $$$x$$$ of a vector space $$$V$$$ is a null vector for a positive sesquilinear form $$$\mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{}$$$ on $$$V$$$ if $$$\mathopen{}\left\langle{}x, x\right\rangle\mathclose{}= 0$$$.

Exercise I.11

Let $$$\mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{}$$$ be a positive sesquilinear form on a vector space $$$V$$$. Prove that the set $$$N$$$ of null vectors is a subspace of $$$V$$$; that is, show that $$$\mathbb{C}N= N$$$ and $$$N+N= N$$$.

Hint

Use Proposition I.4 to show that $$$N+N\subseteq N$$$.

Exercise I.12

Let $$$S$$$ be a subspace of a vector space $$$V$$$. For elements $$$x$$$ and $$$y$$$ of $$$V$$$, write $$$x\equiv y$$$ if $$$x-y$$$ is in $$$S$$$. Prove that this defines an equivalence relation on $$$V$$$.

Definition I.13

For a subspace $$$S$$$ of a vector space $$$V$$$, write $$$x+S$$$ for the equivalence class of the element $$$x$$$ under the equivalence relation of Exercise I.12, and write $$$V/S$$$ for the quotient space, which is the set of all equivalence classes.

Exercise I.14

Let $$$S$$$ be a subspace of a complex vector space $$$V$$$. Prove that $$$V/S$$$ is also a complex vector space.

As we see next, a positive sesquilinear form can give rise to an inner product by modding out the null vectors.

Proposition I.15

Let $$$\mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{}$$$ be a positive sesquilinear form on a vector space $$$V$$$. Let $$$N$$$ be the subspace of null vectors. Then $$$\mathopen{}\left\langle{}x+N, y+N\right\rangle\mathclose{}= \mathopen{}\left\langle{}x, y\right\rangle\mathclose{}$$$ defines an inner product on $$$V/N$$$.

Proof. See Exercise I.16.

Exercise I.16

Prove Proposition I.15 as follows: Using Proposition I.3, show that $$$\mathopen{}\left\langle{}V, N\right\rangle\mathclose{}= 0= \mathopen{}\left\langle{}N, V\right\rangle\mathclose{}$$$. Next, use that $$$\mathopen{}\left\langle{}V, N\right\rangle\mathclose{}= 0= \mathopen{}\left\langle{}N, V\right\rangle\mathclose{}$$$ to show that the new inner product is well-defined. Finally, show that the properties of Definition I.5 are satisfied.

Definition I.17

Let $$$G$$$ be a group. The complex group algebra $$$\mathbb{C}G$$$ is the complex vector space spanned by the set of symbols $$$\mathopen{}\left\{\, {δ}_{g}\,\middle\vert\, g\in G\,\right\}\mathclose{}$$$ with multiplication defined by $$${δ}_{g}{δ}_{h}= {δ}_{gh}$$$.

Example I.18

Let $$$G$$$ be a group and suppose $$$φ : G \to \mathbb{C}$$$ is a function such that for each positive integer $$$k$$$, $$\sum_{i,j=1}^{k}{} {λ}_{i}\overline{{λ}_{j}}φ\mathopen{}\left( {g}_{i}{g}_{j}^{-1}\right)\mathclose{} \geq 0$$ for any complex numbers $$${λ}_{1}$$$, …, $$${λ}_{k}$$$ and group elements $$${g}_{1}$$$, …, $$${g}_{k}$$$ (such a function is called positive definite). You can then define a positive sesquilinear form on $$$\mathbb{C}G$$$ by $$\mathopen{}\left\langle{}\sum_{g\in G}{}{κ}_{g}{δ}_{g}, \sum_{h\in G}{}{λ}_{h}{δ}_{h}\right\rangle\mathclose{}= \sum_{g,h}{} {κ}_{g}\overline{{λ}_{h}}φ\mathopen{}\left( g{h}^{-1}\right)\mathclose{} \text{.}$$ Letting $$$N$$$ be the subspace of null vectors, $$$\mathbb{C}G/N$$$ is a big deal in group representation theory.

Definition I.19

A norm on a vector space $$$V$$$ is a function $$$\mathopen{}\left\lVert{}\cdot\right\rVert\mathclose{} : V \to \mathbb{R}$$$ such that, for all $$$v\in V$$$ and $$$λ\in \mathbb{C}$$$,

1. $$$\mathopen{}\left\lVert{}v\right\rVert\mathclose{}\geq 0$$$.
2. $$$\mathopen{}\left\lVert{}v\right\rVert\mathclose{}= 0$$$ implies $$$v= 0$$$.
3. $$$\mathopen{}\left\lVert{}λv\right\rVert\mathclose{}= \mathopen{}\left\lvert{}λ\right\rvert\mathclose{}\mathopen{}\left\lVert{}v\right\rVert\mathclose{}$$$.
4. $$$\mathopen{}\left\lVert{}v+w\right\rVert\mathclose{}\leq \mathopen{}\left\lVert{}v\right\rVert\mathclose{}+\mathopen{}\left\lVert{}w\right\rVert\mathclose{}$$$.

Definition I.20

A vector space equipped with a norm is a normed linear space.

Proposition I.21

An inner product space with $$$\mathopen{}\left\lVert{}x\right\rVert\mathclose{}= {\mathopen{}\left\langle{}x, x\right\rangle\mathclose{}}^{\frac{1}{2}}$$$ is a normed linear space.

Exercise I.22
Definition I.23

The supremum norm (or infinity norm) of a function $$$f$$$ in $$$\mathrm{C}\mathopen{}\left( \mathopen{}\left[a, b\right]\mathclose{}\right)\mathclose{}$$$ is given by $$$\mathopen{}\left\lVert{}f\right\rVert_\infty\mathclose{}= \max_{t\in \mathopen{}\left[a, b\right]\mathclose{}}{} \mathopen{}\left\lvert{}f\mathopen{}\left( t\right)\mathclose{}\right\rvert\mathclose{}$$$.

Exercise I.24

Prove that the supremum norm is a norm.

Definition I.25

For a real number $$$p$$$ with $$$p\geq 1$$$, the $$$p$$$-norm of a function $$$f$$$ in $$$\mathrm{C}\mathopen{}\left( \mathopen{}\left[a, b\right]\mathclose{}\right)\mathclose{}$$$ is given by $$$\mathopen{}\left\lVert{}f\right\rVert_p\mathclose{}= {\mathopen{}\left(\int _{a}^{b}{}{\mathopen{}\left\lvert{}f\mathopen{}\left( t\right)\mathclose{}\right\rvert\mathclose{}}^{p}\,\mathrm{d}t\right)\mathclose{}}^{\frac{1}{p}}$$$.

Exercise I.26
Exercise I.27