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

## E. Constructions

Proposition I.78

Suppose $$$H$$$ is a Hilbert space, $$${T}_{1}$$$ and $$${T}_{2}$$$ are closed orthogonal subspaces, and $$$T$$$ is a closed subspace such that $$${T}_{1}\subseteq T$$$. Then $$${T}_{1}\oplus {T}_{2}$$$ is closed, and there exists exactly one closed subspace, denoted $$$T\ominus {T}_{1}$$$, such that $$$\mathopen{}\left(T\ominus {T}_{1}\right)\mathclose{}\subseteq T$$$, $$$\mathopen{}\left(T\ominus {T}_{1}\right)\mathclose{}\perp {T}_{1}$$$, and $$$T= {T}_{1}\oplus \mathopen{}\left(T\ominus {T}_{1}\right)\mathclose{}$$$. This subspace is called the orthogonal complement of $$${T}_{1}$$$ with respect to $$$T$$$.

Proof. Exercise.

Exercise I.79

For $$$H$$$ a Hilbert space and $$$T\subseteq H$$$ closed, $$$H\ominus T= {T}^{\perp}$$$.

Definition I.80

The external direct sum of a finite number of Hilbert spaces $$${H}_{1}$$$, …, $$${H}_{n}$$$ with inner products $$$\mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{}_{1}$$$, …, $$$\mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{}_{n}$$$, written $$${H}_{1}\oplus \dotsb\oplus {H}_{n}$$$, is the set of $$$n$$$-tuples $$$\mathopen{}\left\{\, \mathopen{}\left({x}_{1}, \dotsc, {x}_{n}\right)\mathclose{}\,\middle\vert\, , {x}_{i}\in {H}_{i}, \,\right\}\mathclose{}$$$.

Exercise I.81

The direct sum of Hilbert spaces is a Hilbert space with

1. $$$\mathopen{}\left({x}_{1}, \dotsc, {x}_{n}\right)\mathclose{}+\mathopen{}\left({y}_{1}, \dotsc, {y}_{n}\right)\mathclose{}= \mathopen{}\left({x}_{1}+{y}_{1}, \dotsc, {x}_{n}+{y}_{n}\right)\mathclose{}$$$,
2. $$$\mathopen{}\left\langle{}\mathopen{}\left({x}_{1}, \dotsc, {x}_{n}\right)\mathclose{}, \mathopen{}\left({y}_{1}, \dotsc, {y}_{n}\right)\mathclose{}\right\rangle\mathclose{}= \sum_{i=1}^{n}{} \mathopen{}\left\langle{}{x}_{i}, {y}_{i}\right\rangle\mathclose{}_{i}$$$,
3. $$$\mathopen{}\left\lVert{}\mathopen{}\left({x}_{1}, \dotsb, {x}_{n}\right)\mathclose{}\right\rVert\mathclose{}= {\mathopen{}\left(\sum_{i=1}^{n}{} {\mathopen{}\left(\mathopen{}\left\lVert{}{x}_{i}\right\rVert\mathclose{}_{i}\right)\mathclose{}}^{2} \right)\mathclose{}}^{\frac{1}{2}}$$$.

Remark I.82

It makes sense to use the same notation for the external direct sum $$$H= {H}_{1}\oplus \dotsb\oplus {H}_{n}$$$ and the (internal) direct sum $$$K= {H}_{1}\oplus \dotsb\oplus {H}_{n}\subseteq H$$$ since $$$H$$$ is isomorphic to $$$K$$$ in the following way: Let $$${E}_{i}$$$ be the orthogonal projection of $$$K$$$ onto the subspace $$${H}_{i}$$$. Note that $$$\sum_{i=1}^{n}{}{E}_{i}= I$$$. Now define $$$U : K \to H$$$ as $$$x\mapsto \mathopen{}\left({E}_{1}x, \dotsc, {E}_{n}x\right)\mathclose{}$$$. $$$U$$$ is clearly surjective. $$$U$$$ preserves norms since $$${\mathopen{}\left\lVert{}U\mathopen{}\left( x\right)\mathclose{}\right\rVert\mathclose{}}^{2}= \sum_{i=1}^{n}{} {\mathopen{}\left\lVert{}{E}_{i}\mathopen{}\left( x\right)\mathclose{}\right\rVert\mathclose{}}^{2} = {\mathopen{}\left\lVert{}\mathopen{}\left(\sum_{i=1}^{n}{} {E}_{i} \right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}\right\rVert\mathclose{}}^{2}= {\mathopen{}\left\lVert{}x\right\rVert\mathclose{}}^{2}$$$.

Definition I.83

If $$${V}_{1}$$$, …, $$${V}_{n}$$$ are vector spaces, a function on $$${V}_{1}\times \dotsb\times {V}_{n}$$$ is multilinear if it is linear on each $$${V}_{i}$$$.

Theorem I.84

If $$${V}_{1}$$$, …, $$${V}_{n}$$$, $$$V$$$ are vector spaces, there exists a unique (up to isomorphism) vector space $$$W$$$ and a map $$$γ : {V}_{1}\times \dotsb\times {V}_{n} \to W$$$ with the property that if $$$φ : {V}_{1}\times \dotsb\times {V}_{n} \to V$$$ is a multilinear map, there exists a unique linear map $$$ψ : W \to V$$$ such that $$$φ= ψ\circ γ$$$.

Definition I.85

The vector space $$$W$$$ and mapping $$$γ$$$ of Theorem I.84 is called the algebraic tensor product of $$${V}_{1}$$$, …, $$${V}_{n}$$$ and is written $$${V}_{1}\otimes \dotsb\otimes {V}_{n}$$$. If $$${x}_{i}\in {V}_{i}$$$, then $$$γ\mathopen{}\left( \mathopen{}\left({x}_{1}, \dotsc, {x}_{n}\right)\mathclose{}\right)\mathclose{}$$$ is written $$${x}_{1}\otimes \dotsb\otimes {x}_{n}$$$.

Although the simple tensors $$${x}_{1}\otimes \dotsb\otimes {x}_{n}$$$ are the image of $$$γ$$$, they are not the entire tensor product space. Elements of $$${V}_{1}\otimes \dotsb\otimes {V}_{n}$$$, may, however, be regarded as (not always unique) sums of simple tensors.

Remark I.86

Alternately, the algebraic tensor product may be thought of as the quotient space $$$\mathopen{}\left({V}_{1}\times \dotsb\times {V}_{n}\right)\mathclose{}/N$$$ where $$$N$$$ is the subspace spanned by elements of multilinear form.

Exercise I.87

If $$$V$$$ and $$$W$$$ are vector spaces with $$${x}_{1}$$$ and $$${x}_{2}$$$ in $$$V$$$ and with $$${y}_{1}$$$ and $$${y}_{2}$$$ in $$$W$$$, the following hold for $$$λ\in \mathbb{C}$$$:

1. $$$\mathopen{}\left(λ{x}_{1}\right)\mathclose{}\otimes {y}_{1}= {x}_{1}\otimes \mathopen{}\left(λ{y}_{1}\right)\mathclose{}= λ\mathopen{}\left({x}_{1}\otimes {y}_{1}\right)\mathclose{}$$$.
2. $$$\mathopen{}\left({x}_{1}+{x}_{2}\right)\mathclose{}\otimes {y}_{1}= {x}_{1}\otimes {y}_{1}+{x}_{2}\otimes {y}_{1}$$$.
3. $$${x}_{1}\otimes \mathopen{}\left({y}_{1}+{y}_{2}\right)\mathclose{}= {x}_{1}\otimes {y}_{1}+{x}_{1}\otimes {y}_{2}$$$.

In general, if $$$X$$$ and $$$Y$$$ are normed spaces, there are a number of possible norms that may be defined on $$$X\otimes Y$$$ (see Istratescu  and Murphy ). In the case of Hilbert spaces, however, a natural inner product helps simplify the situation.

Theorem I.88

If $$$\mathopen{}\left(H, \mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{}_{H}\right)\mathclose{}$$$ and $$$\mathopen{}\left(K, \mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{}_{K}\right)\mathclose{}$$$ are Hilbert spaces, then $$$H\otimes K$$$ is an inner product space with inner product $$\mathopen{}\left\langle{}{h}_{1}\otimes {k}_{1}, {h}_{2}\otimes {k}_{2}\right\rangle\mathclose{}= \mathopen{}\left\langle{}{h}_{1}, {h}_{2}\right\rangle\mathclose{}_{H}\mathopen{}\left\langle{}{k}_{1}, {k}_{2}\right\rangle\mathclose{}_{K}$$ and induced norm $$\mathopen{}\left\lVert{}{h}_{1}\otimes {k}_{1}\right\rVert\mathclose{}= \mathopen{}\left\lVert{}{h}_{1}\right\rVert\mathclose{}_{H}\mathopen{}\left\lVert{}{k}_{1}\right\rVert\mathclose{}_{K} \text{.}$$

Proof. Follows from the algebraic properties of simple tensors and complex conjugation.

Definition I.89

If $$$H$$$ and $$$K$$$ are Hilbert spaces, the Hilbert space derived by completion of $$$H\otimes K$$$ with respect to the norm induced by the inner product of Theorem Theorem I.88 is called the Hilbert space tensor product of $$$H$$$ and $$$K$$$, and is denoted $$$H\mathbin{\hat{\otimes}} K$$$.

Remark I.90

Sometimes the notation $$$H\odot K$$$ is used for the algebraic tensor product, and $$$H\otimes K$$$ for its completion.

Example I.91

Let $$${μ}_{1}$$$ and $$${μ}_{2}$$$ be measures on $$$\mathbb{R}$$$. The algebraic tensor product $$$\mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathbb{R}, {μ}_{1}\right)\mathclose{}\otimes \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathbb{R}, {μ}_{2}\right)\mathclose{}$$$ is composed of equivalence classes of functions, square integrable on $$${\mathbb{R}}^{2}$$$ with respect to the measure $$${μ}_{1}{μ}_{2}$$$. For $$$f$$$ and $$$g$$$ in $$$\mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathbb{R}, {μ}_{1}\right)\mathclose{}\otimes \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathbb{R}, {μ}_{2}\right)\mathclose{}$$$, $$\mathopen{}\left\langle{}f, g\right\rangle\mathclose{}= \int {} \int {} \overline{ f\mathopen{}\left( x, y\right)\mathclose{} }g\mathopen{}\left( x, y\right)\mathclose{} \,\mathrm{d} {μ}_{1}\mathopen{}\left( x\right)\mathclose{} \,\mathrm{d} {μ}_{2}\mathopen{}\left( y\right)\mathclose{}$$ Since $$$\mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathbb{R}, {μ}_{1}\right)\mathclose{}\otimes \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathbb{R}, {μ}_{2}\right)\mathclose{}$$$ contains all step functions on $$${\mathbb{R}}^{2}$$$, $$\mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathbb{R}, {μ}_{1}\right)\mathclose{}\mathbin{\hat{\otimes}} \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathbb{R}, {μ}_{2}\right)\mathclose{}\simeq \mathrm{L}^{\mathrm{2}}\mathopen{}\left( {\mathbb{R}}^{2}, {μ}_{1}\times {μ}_{2}\right)\mathclose{}\text{.}$$

Exercise I.92

If $$${M}_{H}$$$ and $$${M}_{K}$$$ are dense subspaces of $$$H$$$ and $$$K$$$ respectively, then $$${M}_{H}\otimes {M}_{K}$$$ is dense in $$$H\mathbin{\hat{\otimes}} K$$$.

Exercise I.93

If $$${E}_{H}$$$ and $$${E}_{K}$$$ are orthonormal bases for $$$H$$$ and $$$K$$$ respectively, then $$$\mathopen{}\left\{\, {e}_{H}\otimes {e}_{K}\,\middle\vert\, {e}_{H}\in {E}_{H}, {e}_{K}\in {E}_{K}\,\right\}\mathclose{}$$$ is an orthonormal basis for $$$H\mathbin{\hat{\otimes}} K$$$. What does this say about the dimension of $$$H\mathbin{\hat{\otimes}} K$$$ ?

Exercise I.94

If $$$H$$$ and $$$K$$$ are non-trivial, then $$$H\mathbin{\hat{\otimes}} K$$$ is separable if and only if $$$H$$$ and $$$K$$$ are separable.

Exercise I.95

The inner product space defined in Theorem I.88 is complete if and only if $$$H$$$ or $$$K$$$ is finite dimensional.