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