Lecture Notes in Functional Analysis

by William L. Paschke

edition 0.9




I. Hilbert Space

II. Bounded Operators

III. Compact Operators

IV. The Spectral Theorem

Index and References

E. Constructions

Proposition I.78

Suppose HHilbert space\( H \) is a Hilbert space, Tclosed subspace1one \( {T}_{1} \) and Tclosed subspace2two \( {T}_{2} \) are closed orthogonal subspaces, and Tclosed subspace\( T \) is a closed subspace such that Tclosed subspace1onesubsetTclosed subspace \( {T}_{1}\subseteq T \). Then Tclosed subspace1oneTclosed subspace2two \( {T}_{1}\oplus {T}_{2} \) is closed, and there exists exactly one closed subspace, denoted Tclosed subspacedirect minusTclosed subspace1one \( T\ominus {T}_{1} \), such that (Tclosed subspaceTclosed subspace1one)subsetTclosed subspace \( \mathopen{}\left(T\ominus {T}_{1}\right)\mathclose{}\subseteq T \), (Tclosed subspaceTclosed subspace1one)Tclosed subspace1one \( \mathopen{}\left(T\ominus {T}_{1}\right)\mathclose{}\perp {T}_{1} \), and Tclosed subspace=equalsTclosed subspace1one(Tclosed subspaceTclosed subspace1one) \( T= {T}_{1}\oplus \mathopen{}\left(T\ominus {T}_{1}\right)\mathclose{} \). This subspace is called the orthogonal complement of Tclosed subspace1one \( {T}_{1} \) with respect to Tclosed subspace\( T \).

Proof. Exercise.

Exercise I.79

For HHilbert space\( H \) a Hilbert space and Tclosed subspacesubsetHHilbert space \( T\subseteq H \) closed, HHilbert spaceTclosed subspace=equalsTclosed subspace \( H\ominus T= {T}^{\perp} \).

Definition I.80

The external direct sum of a finite number of Hilbert spaces HHilbert space1one\( {H}_{1} \), …, HHilbert spaceninteger\( {H}_{n} \) with inner products ·, ·1one\( \mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{}_{1} \), …, ·, ·ninteger\( \mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{}_{n} \), written HHilbert space1onedirect sumdirect sumHHilbert spaceninteger \( {H}_{1}\oplus \dotsb\oplus {H}_{n} \), is the set of ninteger\( n \)-tuples {set(tuplexvector1one, , xvectorninteger)tuple|such that xvectoriintegerelement ofHHilbert spaceiinteger }set \( \mathopen{}\left\{\, \mathopen{}\left({x}_{1}, \dotsc, {x}_{n}\right)\mathclose{}\,\middle\vert\, , {x}_{i}\in {H}_{i}, \,\right\}\mathclose{} \).

Exercise I.81

The direct sum of Hilbert spaces is a Hilbert space with

  1. (tuplexvector1one, , xvectorninteger)tuple+plus(tupleyvector1one, , yvectorninteger)tuple=equals(tuplexvector1one+plusyvector1one, , xvectorninteger+plusyvectorninteger)tuple \( \mathopen{}\left({x}_{1}, \dotsc, {x}_{n}\right)\mathclose{}+\mathopen{}\left({y}_{1}, \dotsc, {y}_{n}\right)\mathclose{}= \mathopen{}\left({x}_{1}+{y}_{1}, \dotsc, {x}_{n}+{y}_{n}\right)\mathclose{} \),
  2. (tuplexvector1one, , xvectorninteger)tuple, (tupleyvector1one, , yvectorninteger)tuple=equalssummationiinteger=1oneninteger xvectoriinteger, yvectoriintegeriinteger \( \mathopen{}\left\langle{}\mathopen{}\left({x}_{1}, \dotsc, {x}_{n}\right)\mathclose{}, \mathopen{}\left({y}_{1}, \dotsc, {y}_{n}\right)\mathclose{}\right\rangle\mathclose{}= \sum_{i=1}^{n}{} \mathopen{}\left\langle{}{x}_{i}, {y}_{i}\right\rangle\mathclose{}_{i} \),
  3. (tuplexvector1one, , xvectorninteger)tuple=equals (summationiinteger=1oneninteger (xvectoriintegeriinteger) 2two ) 1one2two \( \mathopen{}\left\lVert{}\mathopen{}\left({x}_{1}, \dotsb, {x}_{n}\right)\mathclose{}\right\rVert\mathclose{}= {\mathopen{}\left(\sum_{i=1}^{n}{} {\mathopen{}\left(\mathopen{}\left\lVert{}{x}_{i}\right\rVert\mathclose{}_{i}\right)\mathclose{}}^{2} \right)\mathclose{}}^{\frac{1}{2}} \).

Remark I.82

It makes sense to use the same notation for the external direct sum HHilbert space=equalsHHilbert space1oneHHilbert spaceninteger \( H= {H}_{1}\oplus \dotsb\oplus {H}_{n} \) and the (internal) direct sum Kcompact operator=equalsHHilbert space1oneHHilbert spacenintegersubsetHHilbert space \( K= {H}_{1}\oplus \dotsb\oplus {H}_{n}\subseteq H \) since HHilbert space\( H \) is isomorphic to Kcompact operator\( K \) in the following way: Let Econvex setiinteger \( {E}_{i} \) be the orthogonal projection of Kcompact operator\( K \) onto the subspace HHilbert spaceiinteger \( {H}_{i} \). Note that summationiinteger=1onenintegerEconvex setiinteger=equalsI \( \sum_{i=1}^{n}{}{E}_{i}= I \). Now define U:mapsKcompact operatortoHHilbert space \( U : K \to H \) as xvectoris mapped to(tupleEconvex set1onetimesxvector, , Econvex setnintegertimesxvector)tuple \( x\mapsto \mathopen{}\left({E}_{1}x, \dotsc, {E}_{n}x\right)\mathclose{} \). U\( U \) is clearly surjective. U\( U \) preserves norms since U(xvector) 2two =equalssummationiinteger=1oneninteger Econvex setiinteger(xvector) 2two =equals (summationiinteger=1oneninteger Econvex setiinteger )(xvector) 2two =equals xvector 2two \( {\mathopen{}\left\lVert{}U\mathopen{}\left( x\right)\mathclose{}\right\rVert\mathclose{}}^{2}= \sum_{i=1}^{n}{} {\mathopen{}\left\lVert{}{E}_{i}\mathopen{}\left( x\right)\mathclose{}\right\rVert\mathclose{}}^{2} = {\mathopen{}\left\lVert{}\mathopen{}\left(\sum_{i=1}^{n}{} {E}_{i} \right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}\right\rVert\mathclose{}}^{2}= {\mathopen{}\left\lVert{}x\right\rVert\mathclose{}}^{2} \).

Definition I.83

If Vvector space1one\( {V}_{1} \), …, Vvector spaceninteger\( {V}_{n} \) are vector spaces, a function on Vvector space1one×Cartesian product×Cartesian productVvector spaceninteger \( {V}_{1}\times \dotsb\times {V}_{n} \) is multilinear if it is linear on each Vvector spaceiinteger \( {V}_{i} \).

Theorem I.84

If Vvector space1one\( {V}_{1} \), …, Vvector spaceninteger\( {V}_{n} \), Vvector space\( V \) are vector spaces, there exists a unique (up to isomorphism) vector space Wvector space\( W \) and a map γmapping:mapsVvector space1one×Cartesian product×Cartesian productVvector spacenintegertoWvector space \( γ : {V}_{1}\times \dotsb\times {V}_{n} \to W \) with the property that if φmultilinear map:mapsVvector space1one×Cartesian product×Cartesian productVvector spacenintegertoVvector space \( φ : {V}_{1}\times \dotsb\times {V}_{n} \to V \) is a multilinear map, there exists a unique linear map ψlinear map:mapsWvector spacetoVvector space \( ψ : W \to V \) such that φmultilinear map=equalsψlinear mapcompositionγmapping \( φ= ψ\circ γ \).

Definition I.85

The vector space Wvector space\( W \) and mapping γmapping\( γ \) of Theorem I.84 is called the algebraic tensor product of Vvector space1one\( {V}_{1} \), …, Vvector spaceninteger\( {V}_{n} \) and is written Vvector space1onetensor producttensor productVvector spaceninteger \( {V}_{1}\otimes \dotsb\otimes {V}_{n} \). If xvectoriintegerelement ofVvector spaceiinteger \( {x}_{i}\in {V}_{i} \), then γmapping((xvector1one, , xvectorninteger)) \( γ\mathopen{}\left( \mathopen{}\left({x}_{1}, \dotsc, {x}_{n}\right)\mathclose{}\right)\mathclose{} \) is written xvector1onexvectorninteger \( {x}_{1}\otimes \dotsb\otimes {x}_{n} \).

Although the simple tensors xvector1onexvectorninteger \( {x}_{1}\otimes \dotsb\otimes {x}_{n} \) are the image of γmapping\( γ \), they are not the entire tensor product space. Elements of Vvector space1oneVvector spaceninteger \( {V}_{1}\otimes \dotsb\otimes {V}_{n} \), 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 (Vvector space1one×Cartesian product×Cartesian productVvector spaceninteger)/Nsubspace \( \mathopen{}\left({V}_{1}\times \dotsb\times {V}_{n}\right)\mathclose{}/N \) where Nsubspace\( N \) is the subspace spanned by elements of multilinear form.

Exercise I.87

If Vvector space\( V \) and Wvector space\( W \) are vector spaces with xvector1one \( {x}_{1} \) and xvector2two \( {x}_{2} \) in Vvector space\( V \) and with yvector1one \( {y}_{1} \) and yvector2two \( {y}_{2} \) in Wvector space\( W \), the following hold for λcomplex numberelement ofCcomplex numbers \( λ\in \mathbb{C} \):

  1. (λcomplex numbertimesxvector1one)yvector1one=equalsxvector1one(λcomplex numbertimesyvector1one)=equalsλcomplex numbertimes(xvector1oneyvector1one) \( \mathopen{}\left(λ{x}_{1}\right)\mathclose{}\otimes {y}_{1}= {x}_{1}\otimes \mathopen{}\left(λ{y}_{1}\right)\mathclose{}= λ\mathopen{}\left({x}_{1}\otimes {y}_{1}\right)\mathclose{} \).
  2. (xvector1one+plusxvector2two)yvector1one=equalsxvector1oneyvector1one+plusxvector2twoyvector1one \( \mathopen{}\left({x}_{1}+{x}_{2}\right)\mathclose{}\otimes {y}_{1}= {x}_{1}\otimes {y}_{1}+{x}_{2}\otimes {y}_{1} \).
  3. xvector1one(yvector1one+plusyvector2two)=equalsxvector1oneyvector1one+plusxvector1oneyvector2two \( {x}_{1}\otimes \mathopen{}\left({y}_{1}+{y}_{2}\right)\mathclose{}= {x}_{1}\otimes {y}_{1}+{x}_{1}\otimes {y}_{2} \).

In general, if Xnormed linear space\( X \) and Ynormed linear space\( Y \) are normed spaces, there are a number of possible norms that may be defined on Xnormed linear spaceYnormed linear space \( X\otimes Y \) (see Istratescu [4] and Murphy [8]). In the case of Hilbert spaces, however, a natural inner product helps simplify the situation.

Theorem I.88

If (tupleHHilbert space, ·, ·HHilbert space)tuple \( \mathopen{}\left(H, \mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{}_{H}\right)\mathclose{} \) and (tupleKHilbert space, ·, ·KHilbert space)tuple \( \mathopen{}\left(K, \mathopen{}\left\langle{}\cdot, \cdot\right\rangle\mathclose{}_{K}\right)\mathclose{} \) are Hilbert spaces, then HHilbert spaceKHilbert space \( H\otimes K \) is an inner product space with inner product hvector1onekvector1one, hvector2twokvector2two=equalshvector1one, hvector2twoHHilbert spacetimeskvector1one, kvector2twoKHilbert space \[ \mathopen{}\left\langle{}{h}_{1}\otimes {k}_{1}, {h}_{2}\otimes {k}_{2}\right\rangle\mathclose{}= \mathopen{}\left\langle{}{h}_{1}, {h}_{2}\right\rangle\mathclose{}_{H}\mathopen{}\left\langle{}{k}_{1}, {k}_{2}\right\rangle\mathclose{}_{K} \] and induced norm hvector1onekvector1one=equalshvector1oneHHilbert spacetimeskvector1oneKHilbert space . \[ \mathopen{}\left\lVert{}{h}_{1}\otimes {k}_{1}\right\rVert\mathclose{}= \mathopen{}\left\lVert{}{h}_{1}\right\rVert\mathclose{}_{H}\mathopen{}\left\lVert{}{k}_{1}\right\rVert\mathclose{}_{K} \text{.} \]

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

Definition I.89

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

Remark I.90

Sometimes the notation HHilbert spacealgebraic tensor productKHilbert space \( H\odot K \) is used for the algebraic tensor product, and HHilbert spaceKHilbert space \( H\otimes K \) for its completion.

Example I.91

Let μmeasure1one \( {μ}_{1} \) and μmeasure2two \( {μ}_{2} \) be measures on Rreal numbers\( \mathbb{R} \). The algebraic tensor product L2Lebesgue space(Rreal numbersμmeasure1one)L2Lebesgue space(Rreal numbersμmeasure2two) \( \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathbb{R}, {μ}_{1}\right)\mathclose{}\otimes \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathbb{R}, {μ}_{2}\right)\mathclose{} \) is composed of equivalence classes of functions, square integrable on Rreal numbers2two \( {\mathbb{R}}^{2} \) with respect to the measure μmeasure1onetimesμmeasure2two \( {μ}_{1}{μ}_{2} \). For ffunction\( f \) and ggroup element\( g \) in L2Lebesgue space(Rreal numbersμmeasure1one)L2Lebesgue space(Rreal numbersμmeasure2two) \( \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathbb{R}, {μ}_{1}\right)\mathclose{}\otimes \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathbb{R}, {μ}_{2}\right)\mathclose{} \), ffunction, ggroup element=equalsintegral integral ffunction(xvectoryvector) ¯complex conjugatetimesggroup element(xvectoryvector) d μmeasure1one(xvector) d μmeasure2two(yvector) \[ \mathopen{}\left\langle{}f, g\right\rangle\mathclose{}= \int {} \int {} \overline{ f\mathopen{}\left( x, y\right)\mathclose{} }g\mathopen{}\left( x, y\right)\mathclose{} \,\mathrm{d} {μ}_{1}\mathopen{}\left( x\right)\mathclose{} \,\mathrm{d} {μ}_{2}\mathopen{}\left( y\right)\mathclose{} \] Since L2Lebesgue space(Rreal numbersμmeasure1one)L2Lebesgue space(Rreal numbersμmeasure2two) \( \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathbb{R}, {μ}_{1}\right)\mathclose{}\otimes \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathbb{R}, {μ}_{2}\right)\mathclose{} \) contains all step functions on Rreal numbers2two \( {\mathbb{R}}^{2} \), L2Lebesgue space(Rreal numbersμmeasure1one)L2Lebesgue space(Rreal numbersμmeasure2two)L2Lebesgue space(Rreal numbers2twoμmeasure1one×Cartesian productμmeasure2two). \[ \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathbb{R}, {μ}_{1}\right)\mathclose{}\mathbin{\hat{\otimes}} \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathbb{R}, {μ}_{2}\right)\mathclose{}\simeq \mathrm{L}^{\mathrm{2}}\mathopen{}\left( {\mathbb{R}}^{2}, {μ}_{1}\times {μ}_{2}\right)\mathclose{}\text{.} \]

Exercise I.92

If MmatrixHHilbert space \( {M}_{H} \) and MmatrixKHilbert space \( {M}_{K} \) are dense subspaces of HHilbert space\( H \) and KHilbert space\( K \) respectively, then MmatrixHHilbert spaceMmatrixKHilbert space \( {M}_{H}\otimes {M}_{K} \) is dense in HHilbert spaceKHilbert space \( H\mathbin{\hat{\otimes}} K \).

Exercise I.93

If Eorthonormal basisHHilbert space \( {E}_{H} \) and Eorthonormal basisKHilbert space \( {E}_{K} \) are orthonormal bases for HHilbert space\( H \) and KHilbert space\( K \) respectively, then {seteunit vectorHHilbert spaceeunit vectorKHilbert space|such thateunit vectorHHilbert spaceelement ofEorthonormal basisHHilbert spaceeunit vectorKHilbert spaceelement ofEorthonormal basisKHilbert space}set \( \mathopen{}\left\{\, {e}_{H}\otimes {e}_{K}\,\middle\vert\, {e}_{H}\in {E}_{H}, {e}_{K}\in {E}_{K}\,\right\}\mathclose{} \) is an orthonormal basis for HHilbert spaceKHilbert space \( H\mathbin{\hat{\otimes}} K \). What does this say about the dimension of HHilbert spaceKHilbert space \( H\mathbin{\hat{\otimes}} K \) ?

Exercise I.94

If HHilbert space\( H \) and KHilbert space\( K \) are non-trivial, then HHilbert spaceKHilbert space \( H\mathbin{\hat{\otimes}} K \) is separable if and only if HHilbert space\( H \) and KHilbert space\( K \) are separable.

Exercise I.95

The inner product space defined in Theorem I.88 is complete if and only if HHilbert space\( H \) or KHilbert space\( K \) is finite dimensional.

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