Lecture Notes in Functional Analysis

by William L. Paschke

edition 0.9

image/svg+xml

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 Sset\( S \) of an inner product space Vinner product space\( V \) is called orthonormal provided xvector=equals1one\( \mathopen{}\left\lVert{}x\right\rVert\mathclose{}= 1 \) and xvectoryvector\( x\perp y \) for all xvector\( x \) and yvector\( y \) in Sset\( S \).

Proposition I.69

Let Sset=equals{seteunit vector1oneeunit vectorninteger}set\( S= \mathopen{}\left\{\, {e}_{1}, \dotsc, {e}_{n}\,\right\}\mathclose{} \) be an orthonormal set in an inner product space Vinner product space\( V \). For complex numbers αcomplex number1one\( {α}_{1} \), …, αcomplex numberninteger\( {α}_{n} \) and βcomplex number1one\( {β}_{1} \), …, βcomplex numberninteger\( {β}_{n} \), summationiinteger=1onenintegerαcomplex numberiintegertimeseunit vectoriinteger, summationjinteger=1onenintegerβcomplex numberjintegertimeseunit vectorjinteger=equalssummationiinteger=1onenintegerαcomplex numberiintegertimesβcomplex numberiinteger¯complex conjugate . \[ \mathopen{}\left\langle{}\sum_{i=1}^{n}{}{α}_{i}{e}_{i}, \sum_{j=1}^{n}{}{β}_{j}{e}_{j}\right\rangle\mathclose{}= \sum_{i=1}^{n}{}{α}_{i}\overline{{β}_{i}} \text{.} \]

Proof. Note that eunit vectoriinteger, eunit vectorjinteger=equals0zero\( \mathopen{}\left\langle{}{e}_{i}, {e}_{j}\right\rangle\mathclose{}= 0 \) for iintegernot equal tojinteger\( i\neq j \) and eunit vectoriinteger, eunit vectoriinteger=equals1one\( \mathopen{}\left\langle{}{e}_{i}, {e}_{i}\right\rangle\mathclose{}= 1 \).

Proposition I.70

Let Sset=equals{seteunit vector1oneeunit vectorninteger}set\( S= \mathopen{}\left\{\, {e}_{1}, \dotsc, {e}_{n}\,\right\}\mathclose{} \) be an orthonormal set in an inner product space Vinner product space\( V \). Let Wvector space=equalsspanspan(Sset)\( W= \operatorname{span}\mathopen{}\left( S\right)\mathclose{} \). Define a linear map Qlinear map:mapsVinner product spacetoWvector space \( Q : V \to W \) by Qlinear map(xvector)=equalssummationiinteger=1onenintegerxvector, eunit vectoriintegertimeseunit vectoriinteger . \[ Q\mathopen{}\left( x\right)\mathclose{}= \sum_{i=1}^{n}{}\mathopen{}\left\langle{}x, {e}_{i}\right\rangle\mathclose{}{e}_{i} \text{.} \] Then, for all xvectorelement ofVinner product space\( x\in V \), xvector-minusQlinear map(xvector)element ofWvector space\( x-Q\mathopen{}\left( x\right)\mathclose{}\in {W}^{\perp} \), Qlinear map(xvector)less than or equal toxvector\( \mathopen{}\left\lVert{}Q\mathopen{}\left( x\right)\mathclose{}\right\rVert\mathclose{}\leq \mathopen{}\left\lVert{}x\right\rVert\mathclose{} \), xvector-minusQlinear map(xvector)less than or equal toxvector\( \mathopen{}\left\lVert{}x-Q\mathopen{}\left( x\right)\mathclose{}\right\rVert\mathclose{}\leq \mathopen{}\left\lVert{}x\right\rVert\mathclose{} \), and Qlinear map(xvector)\( Q\mathopen{}\left( x\right)\mathclose{} \) is the closest point in Wvector space\( W \) to xvector\( x \).

Proof. To prove the first claim, note that, for all jinteger\( j \), Qlinear map(xvector), eunit vectorjinteger=equalsxvector, eunit vectorjinteger \( \mathopen{}\left\langle{}Q\mathopen{}\left( x\right)\mathclose{}, {e}_{j}\right\rangle\mathclose{}= \mathopen{}\left\langle{}x, {e}_{j}\right\rangle\mathclose{} \). Next, for the last claim, we can write xvector=equalsQlinear map(xvector)+plus(xvector-minusQlinear map(xvector))\( x= Q\mathopen{}\left( x\right)\mathclose{}+\mathopen{}\left(x-Q\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{} \) with Qlinear map(xvector)element ofWvector space\( Q\mathopen{}\left( x\right)\mathclose{}\in W \) and xvector-minusQlinear map(xvector)element ofWvector space\( x-Q\mathopen{}\left( x\right)\mathclose{}\in {W}^{\perp} \). Then xvector2two=equalsQlinear map(xvector)2two+plusxvector-minusQlinear map(xvector)2two \( {\mathopen{}\left\lVert{}x\right\rVert\mathclose{}}^{2}= {\mathopen{}\left\lVert{}Q\mathopen{}\left( x\right)\mathclose{}\right\rVert\mathclose{}}^{2}+{\mathopen{}\left\lVert{}x-Q\mathopen{}\left( x\right)\mathclose{}\right\rVert\mathclose{}}^{2} \). Finally, for any wvectorelement ofWvector space\( w\in W \), xvector-minuswvectorgreater than or equal to(xvector-minuswvector)-minus(Qlinear map(xvector-minuswvector))=equalsxvector-minuswvector-minusQlinear map(xvector)+plusQlinear map(wvector) \( \mathopen{}\left\lVert{}x-w\right\rVert\mathclose{}\geq \mathopen{}\left\lVert{}\mathopen{}\left(x-w\right)\mathclose{}-\mathopen{}\left(Q\mathopen{}\left( x-w\right)\mathclose{}\right)\mathclose{}\right\rVert\mathclose{}= \mathopen{}\left\lVert{}x-w-Q\mathopen{}\left( x\right)\mathclose{}+Q\mathopen{}\left( w\right)\mathclose{}\right\rVert\mathclose{} \). Since wvectorelement ofWvector space\( w\in W \), Qlinear map(wvector)=equalswvector\( Q\mathopen{}\left( w\right)\mathclose{}= w \).

Proposition I.71 (Gram-Schmidt Procedure)

Let xvector1one\( {x}_{1} \), xvector2two\( {x}_{2} \), … be a linearly independent sequence in an inner product space Vinner product space\( V \). Then there exists an orthonormal sequence eunit vector1one\( {e}_{1} \), eunit vector2two\( {e}_{2} \), …, in Vinner product space\( V \) such that spanspan({setxvector1onexvector2two}set)=equalsspanspan({seteunit vector1oneeunit vector2two}set)\( \operatorname{span}\mathopen{}\left( \mathopen{}\left\{\, {x}_{1}, {x}_{2}, \dotsc\,\right\}\mathclose{}\right)\mathclose{}= \operatorname{span}\mathopen{}\left( \mathopen{}\left\{\, {e}_{1}, {e}_{2}, \dotsc\,\right\}\mathclose{}\right)\mathclose{} \).

Proof. Start with eunit vector1one=equalsxvector1onexvector1one \( {e}_{1}= \frac{{x}_{1}}{\mathopen{}\left\lVert{}{x}_{1}\right\rVert\mathclose{}} \). Inductive step: Let yvectorninteger+plus1one=equalsxvectorninteger+plus1one-minussummationiinteger=1onenintegerxvectorninteger+plus1one, eunit vectoriintegertimeseunit vectoriinteger \[ {y}_{n+1}= {x}_{n+1}-\sum_{i=1}^{n}{}\mathopen{}\left\langle{}{x}_{n+1}, {e}_{i}\right\rangle\mathclose{}{e}_{i} \] (which is not zero by the inductive hypothesis). Then let eunit vectorninteger+plus1one=equalsyvectorninteger+plus1oneyvectorninteger+plus1one \( {e}_{n+1}= \frac{{y}_{n+1}}{\mathopen{}\left\lVert{}{y}_{n+1}\right\rVert\mathclose{}} \).

Proposition I.72 (Bessel's Inequality)

Let eunit vector1one\( {e}_{1} \), eunit vector2two\( {e}_{2} \), … be an orthonormal sequence in an inner product space Vinner product space\( V \). Then summationiinteger=1oneinfinity|modulusxvector, eunit vectoriinteger|modulus2twoless than or equal toxvector2two \[ \sum_{i=1}^{\infty}{}{\mathopen{}\left\lvert{}\mathopen{}\left\langle{}x, {e}_{i}\right\rangle\mathclose{}\right\rvert\mathclose{}}^{2}\leq {\mathopen{}\left\lVert{}x\right\rVert\mathclose{}}^{2} \] for all xvector\( x \) in Vinner product space\( V \). If Vinner product space\( V \) is complete, then summationiinteger=1oneinfinityxvector, eunit vectoriintegertimeseunit vectoriinteger \( \sum_{i=1}^{\infty}{}\mathopen{}\left\langle{}x, {e}_{i}\right\rangle\mathclose{}{e}_{i} \) converges and sums to the orthogonal projection of xvector\( x \) on spanspan({seteunit vector1oneeunit vector2two}set)¯\( \overline{\operatorname{span}\mathopen{}\left( \mathopen{}\left\{\, {e}_{1}, {e}_{2}, \dotsc\,\right\}\mathclose{}\right)\mathclose{}} \).

Proof. The first claim follows from Proposition I.70 since summationninteger=1oneNinteger|modulusxvector, eunit vectorninteger|modulus2twoless than or equal toxvector2two \[ \sum_{n=1}^{N}{}{\mathopen{}\left\lvert{}\mathopen{}\left\langle{}x, {e}_{n}\right\rangle\mathclose{}\right\rvert\mathclose{}}^{2}\leq {\mathopen{}\left\lVert{}x\right\rVert\mathclose{}}^{2} \] for all Ninteger\( N \).

For the second claim, assume Vinner product space=equalsHHilbert space\( V= H \) is a Hilbert space. For Mreal number>greater thanNinteger\( M\gt N \), we have summationninteger=NintegerMreal numberxvector, eunit vectornintegertimeseunit vectorninteger 2two =equalssummationninteger=NintegerMreal number|modulusxvector, eunit vectorninteger|modulus2two , \[ {\mathopen{}\left\lVert{}\sum_{n=N}^{M}{}\mathopen{}\left\langle{}x, {e}_{n}\right\rangle\mathclose{}{e}_{n}\right\rVert\mathclose{}}^{2}= \sum_{n=N}^{M}{}{\mathopen{}\left\lvert{}\mathopen{}\left\langle{}x, {e}_{n}\right\rangle\mathclose{}\right\rvert\mathclose{}}^{2} \text{,} \] which is small for large Ninteger\( N \) and Mreal number\( M \) by the first part. So, the sequence of partial sums for the whole series, summationninteger=1oneinfinityxvector, eunit vectornintegertimeseunit vectorninteger \( \sum_{n=1}^{\infty}{}\mathopen{}\left\langle{}x, {e}_{n}\right\rangle\mathclose{}{e}_{n} \), is Cauchy, and so the series converges, say to wvector\( w \) in HHilbert space\( H \). Then wvector=equalssummationninteger=1oneinfinityxvector, eunit vectornintegertimeseunit vectorninteger \( w= \sum_{n=1}^{\infty}{}\mathopen{}\left\langle{}x, {e}_{n}\right\rangle\mathclose{}{e}_{n} \) and wvector, eunit vectorjinteger=equalssummationninteger=1oneinfinity xvector, eunit vectornintegertimeseunit vectorninteger, eunit vectorjinteger \[ \mathopen{}\left\langle{}w, {e}_{j}\right\rangle\mathclose{}= \sum_{n=1}^{\infty}{} \mathopen{}\left\langle{}x, {e}_{n}\right\rangle\mathclose{}\mathopen{}\left\langle{}{e}_{n}, {e}_{j}\right\rangle\mathclose{} \] (continuity of ·, eunit vectorjinteger \( \mathopen{}\left\langle{}\cdot, {e}_{j}\right\rangle\mathclose{} \) comes from Proposition I.3). So xvector-minuswvectoreunit vectorjinteger\( x-w\perp {e}_{j} \) and xvector-minuswvectorspanspan({seteunit vector1oneeunit vector2two}set) \( x-w\perp \operatorname{span}\mathopen{}\left( \mathopen{}\left\{\, {e}_{1}, {e}_{2}, \dotsc\,\right\}\mathclose{}\right)\mathclose{} \). Further, wvector\( w \) is in spanspan({seteunit vector1oneeunit vector2two}set)¯\( \overline{\operatorname{span}\mathopen{}\left( \mathopen{}\left\{\, {e}_{1}, {e}_{2}, \dotsc\,\right\}\mathclose{}\right)\mathclose{}} \).

Theorem I.73

Let HHilbert space\( H \) be a separable infinite-dimensional Hilbert space. Then there exists an orthonormal set {seteunit vector1oneeunit vector2two}set\( \mathopen{}\left\{\, {e}_{1}, {e}_{2}, \dotsc\,\right\}\mathclose{} \) with spanspan({seteunit vector1oneeunit vector2two}set)¯=equalsHHilbert space \( \overline{\operatorname{span}\mathopen{}\left( \mathopen{}\left\{\, {e}_{1}, {e}_{2}, \dotsc\,\right\}\mathclose{}\right)\mathclose{}}= H \). Further, xvector=equalssummationiinteger=1oneinfinityxvector, eunit vectoriintegertimeseunit vectoriinteger \[ x= \sum_{i=1}^{\infty}{}\mathopen{}\left\langle{}x, {e}_{i}\right\rangle\mathclose{}{e}_{i} \] for all xvectorelement ofHHilbert space\( x\in H \) and xvector, yvector=equalssummationiinteger=1oneinfinityxvector, eunit vectoriintegertimeseunit vectoriinteger, yvector \[ \mathopen{}\left\langle{}x, y\right\rangle\mathclose{}= \sum_{i=1}^{\infty}{}\mathopen{}\left\langle{}x, {e}_{i}\right\rangle\mathclose{}\mathopen{}\left\langle{}{e}_{i}, y\right\rangle\mathclose{} \] for all xvectorelement ofHHilbert space\( x\in H \) and yvectorelement ofHHilbert space\( y\in H \).

Proof. From the hypothesis and the size of HHilbert space\( H \), we get a dense set {setxvector1onexvector2two}set\( \mathopen{}\left\{\, {x}_{1}, {x}_{2}, \dotsc\,\right\}\mathclose{} \) in HHilbert space\( H \) and from it a linearly independent sequence {setwvector1onewvector2two}set\( \mathopen{}\left\{\, {w}_{1}, {w}_{2}, \dotsc\,\right\}\mathclose{} \) whose linear span is dense in HHilbert space\( H \). Apply Proposition I.71 to get an orthonormal sequence {seteunit vector1oneeunit vector2two}set\( \mathopen{}\left\{\, {e}_{1}, {e}_{2}, \dotsc\,\right\}\mathclose{} \) with the same closed span, namely HHilbert space\( H \).

The first claim follows from Proposition I.72. For the second claim, let xvectorninteger=equalssummationiinteger=1onenintegerxvector, eunit vectoriintegertimeseunit vectoriinteger \( {x}_{n}= \sum_{i=1}^{n}{}\mathopen{}\left\langle{}x, {e}_{i}\right\rangle\mathclose{}{e}_{i} \) and yvectorninteger=equalssummationjinteger=1oneinfinityyvector, eunit vectorjintegertimeseunit vectorjinteger, yvector \( {y}_{n}= \sum_{j=1}^{\infty}{}\mathopen{}\left\langle{}y, {e}_{j}\right\rangle\mathclose{}\mathopen{}\left\langle{}{e}_{j}, y\right\rangle\mathclose{} \). So, xvectorninteger, yvectorninteger=equalssummationiinteger=1onenintegerxvector, eunit vectoriintegertimeseunit vectoriinteger, yvector \( \mathopen{}\left\langle{}{x}_{n}, {y}_{n}\right\rangle\mathclose{}= \sum_{i=1}^{n}{}\mathopen{}\left\langle{}x, {e}_{i}\right\rangle\mathclose{}\mathopen{}\left\langle{}{e}_{i}, y\right\rangle\mathclose{} \) by Proposition I.70. Let ninteger\( n \) go to infinity\( \infty \) and notice xvectorninteger, yvectorninteger \( \mathopen{}\left\langle{}{x}_{n}, {y}_{n}\right\rangle\mathclose{} \) converges to xvector, yvector\( \mathopen{}\left\langle{}x, y\right\rangle\mathclose{} \) by the continuity of the inner product, which comes from the estimate |modulusxvectorninteger, yvectorninteger-minusxvector, yvector|modulusless than or equal to|modulusxvectorninteger-minusxvector, yvectorninteger|modulus+plus|modulusxvector, yvectorninteger-minusyvector|modulusless than or equal toxvectorninteger-minusxvectortimesyvectorninteger+plusxvectortimesyvectorninteger-minusyvector. \[ \mathopen{}\left\lvert{}\mathopen{}\left\langle{}{x}_{n}, {y}_{n}\right\rangle\mathclose{}-\mathopen{}\left\langle{}x, y\right\rangle\mathclose{}\right\rvert\mathclose{}\leq \mathopen{}\left\lvert{}\mathopen{}\left\langle{}{x}_{n}-x, {y}_{n}\right\rangle\mathclose{}\right\rvert\mathclose{}+\mathopen{}\left\lvert{}\mathopen{}\left\langle{}x, {y}_{n}-y\right\rangle\mathclose{}\right\rvert\mathclose{}\leq \mathopen{}\left\lVert{}{x}_{n}-x\right\rVert\mathclose{}\mathopen{}\left\lVert{}{y}_{n}\right\rVert\mathclose{}+\mathopen{}\left\lVert{}x\right\rVert\mathclose{}\mathopen{}\left\lVert{}{y}_{n}-y\right\rVert\mathclose{}\text{.} \]

Definition I.74

Let HHilbert space\( H \) be a separable infinite-dimensional Hilbert space. An orthonormal set {seteunit vector1oneeunit vector2two}set\( \mathopen{}\left\{\, {e}_{1}, {e}_{2}, \dotsc\,\right\}\mathclose{} \) such that spanspan({seteunit vector1oneeunit vector2two}set)¯=equalsHHilbert space \( \overline{\operatorname{span}\mathopen{}\left( \mathopen{}\left\{\, {e}_{1}, {e}_{2}, \dotsc\,\right\}\mathclose{}\right)\mathclose{}}= H \) is an orthonormal basis of HHilbert space\( H \).

Definition I.75

An isomorphism of Hilbert spaces HHilbert spaceisomorphicKHilbert space\( H\simeq K \) is an inner-product-preserving linear surjection.

Theorem I.76

If HHilbert space\( H \) is a separable, infinite dimensional Hilbert space with orthonormal basis {seteunit vector1oneeunit vector2two}set\( \mathopen{}\left\{\, {e}_{1}, {e}_{2}, \dotsc\,\right\}\mathclose{} \), then Uisomorphism:mapsHHilbert spacetol2 \( U : H \to \mathrm{l}^{0} \) defined by Uisomorphism(xvector)=equals(sequencexvector, eunit vectorninteger)sequenceninteger=1oneinfinity \( U\mathopen{}\left( x\right)\mathclose{}= \mathopen{}\left(\mathopen{}\left\langle{}x, {e}_{n}\right\rangle\mathclose{}\right)\mathclose{}_{n=1}^{\infty} \) is an isomorphism.

Proof. Notice that Uisomorphism(xvector), Uisomorphism(yvector)=equalsxvector, yvector \( \mathopen{}\left\langle{}U\mathopen{}\left( x\right)\mathclose{}, U\mathopen{}\left( y\right)\mathclose{}\right\rangle\mathclose{}= \mathopen{}\left\langle{}x, y\right\rangle\mathclose{} \) by Theorem I.73. Furthermore, Uisomorphism\( U \) is onto because (sequenceαcomplex numberninteger)sequenceninteger=1oneinfinityelement ofl2 \( \mathopen{}\left({α}_{n}\right)\mathclose{}_{n=1}^{\infty}\in \mathrm{l}^{0} \) makes summationninteger=1oneinfinityαcomplex numbernintegertimeseunit vectorninteger \( \sum_{n=1}^{\infty}{}{α}_{n}{e}_{n} \) converge in HHilbert space\( H \). Since summationninteger=NintegerMintegerαcomplex numbernintegertimeseunit vectorninteger 2two =equalssummationninteger=NintegerMinteger|modulusαcomplex numberninteger|modulus2two , \[ {\mathopen{}\left\lVert{}\sum_{n=N}^{M}{}{α}_{n}{e}_{n}\right\rVert\mathclose{}}^{2}= \sum_{n=N}^{M}{}{\mathopen{}\left\lvert{}{α}_{n}\right\rvert\mathclose{}}^{2} \text{,} \] the sequence of partial sums is Cauchy. Thus we have Uisomorphism(summationninteger=1oneinfinityαcomplex numbernintegertimeseunit vectorninteger)=equals(sequenceαcomplex numberninteger)sequenceninteger=1oneinfinity \( U\mathopen{}\left( \sum_{n=1}^{\infty}{}{α}_{n}{e}_{n}\right)\mathclose{}= \mathopen{}\left({α}_{n}\right)\mathclose{}_{n=1}^{\infty} \).

Example I.77

Let's look at the sine series in L2Lebesgue space([interval0zero, 1one]interval)\( \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{} \). Consider esine functionninteger(treal number)=equals2twotimessinsine(nintegertimesπpitimestreal number) \( {e}_{n}\mathopen{}\left( t\right)\mathclose{}= \sqrt{2}\sin\mathopen{}\left( n\mathrm{\pi}t\right)\mathclose{} \) for nintegerelement ofNnatural numbers (including zero)\( n\in \mathbb{N} \). From calculus its easy to see that {setesine function1oneesine function2two}set\( \mathopen{}\left\{\, {e}_{1}, {e}_{2}, \dotsc\,\right\}\mathclose{} \) is orthonormal. We claim that it is in fact dense in L2Lebesgue space([interval0zero, 1one]interval)\( \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{} \). Take ffunctionelement ofL2Lebesgue space([interval0zero, 1one]interval) \( f\in \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{} \) such that ffunction, esine functionninteger=equals0zero\( \mathopen{}\left\langle{}f, {e}_{n}\right\rangle\mathclose{}= 0 \) for all ninteger\( n \). Extend ffunction\( f \) to [interval1one, 1one]interval\( \mathopen{}\left[{-}1, 1\right]\mathclose{} \) by ffunction(treal number)=equalsffunction(treal number) \( f\mathopen{}\left( {-}t\right)\mathclose{}= {-}f\mathopen{}\left( t\right)\mathclose{} \) for 0zero<less thantreal number<less than1one\( 0\lt t\lt 1 \). Consider also the function gfunction\( g \) defined (almost everywhere) on the unit circle Tcomplex unit circle=equals{setzcomplex number of modulus oneelement ofCcomplex numbers|such that|moduluszcomplex number of modulus one|modulus=equals1one}set \( \mathbb{T}= \mathopen{}\left\{\, z\in \mathbb{C}\,\middle\vert\, \mathopen{}\left\lvert{}z\right\rvert\mathclose{}= 1\,\right\}\mathclose{} \) by gfunction(eEuler's constantπpitimesiimaginary unittimesθcomplex number of modulus one)=equalsffunction(θcomplex number of modulus one) \( g\mathopen{}\left( {\mathrm{e}}^{\mathrm{\pi}\mathrm{i}θ}\right)\mathclose{}= f\mathopen{}\left( θ\right)\mathclose{} \). Notice integralTgfunction(zcomplex number of modulus one)timeszcomplex number of modulus onenintegerdzcomplex number of modulus one=equalsintegral1one1oneffunction(θcomplex number of modulus one)timeseEuler's constantπpitimesiimaginary unittimesnintegertimesθcomplex number of modulus onedθcomplex number of modulus one=equalsintegral1one1one ffunction(θcomplex number of modulus one)times(coscosine(nintegertimesπpitimesθcomplex number of modulus one)+plusiimaginary unittimessinsine(nintegertimesπpitimesθcomplex number of modulus one)) dθcomplex number of modulus one=equals0zero \[ \int _{\mathbb{T}}{}g\mathopen{}\left( z\right)\mathclose{}{z}^{n}\,\mathrm{d}z= \int _{{-}1}^{1}{}f\mathopen{}\left( θ\right)\mathclose{}{\mathrm{e}}^{\mathrm{\pi}\mathrm{i}nθ}\,\mathrm{d}θ= \int _{{-}1}^{1}{} f\mathopen{}\left( θ\right)\mathclose{}\mathopen{}\left(\cos\mathopen{}\left( n\mathrm{\pi}θ\right)\mathclose{}+\mathrm{i}\sin\mathopen{}\left( n\mathrm{\pi}θ\right)\mathclose{}\right)\mathclose{} \,\mathrm{d}θ= 0 \] because ffunction\( f \) is an odd function and ffunction, esine functionninteger=equals0zero\( \mathopen{}\left\langle{}f, {e}_{n}\right\rangle\mathclose{}= 0 \) for all ninteger\( n \).

So we have that integralTgfunction(zcomplex number of modulus one)timesφfunction(zcomplex number of modulus one)dzcomplex number of modulus one=equals0zero \( \int _{\mathbb{T}}{}g\mathopen{}\left( z\right)\mathclose{}φ\mathopen{}\left( z\right)\mathclose{}\,\mathrm{d}z= 0 \) for all φfunctionelement ofCspace of continuous functions(T)\( φ\in \mathrm{C}\mathopen{}\left( \mathbb{T}\right)\mathclose{} \) because polynomials in zcomplex number of modulus one\( z \) and zcomplex number of modulus one1inverse\( {z}^{-1} \) are uniformly dense in Cspace of continuous functions(T)\( \mathrm{C}\mathopen{}\left( \mathbb{T}\right)\mathclose{} \) by the Stone-Weierstrass Theorem and gfunction=equalsgfunctiontimes1identity function\( g= g\mathbf{1} \) in L1Lebesgue space(T)\( \mathrm{L}^{\mathrm{1}}\mathopen{}\left( \mathbb{T}\right)\mathclose{} \). If a sequence (sequenceφfunctionninteger)sequenceninteger=1oneinfinity \( \mathopen{}\left({φ}_{n}\right)\mathclose{}_{n=1}^{\infty} \) converges to φfunction\( φ \) uniformly, then |modulusintegralTgfunction(zcomplex number of modulus one)timesφfunctionninteger(zcomplex number of modulus one)dzcomplex number of modulus one-minusintegralTgfunction(zcomplex number of modulus one)timesφfunction(zcomplex number of modulus one)dzcomplex number of modulus one|modulusless than or equal togfunctiontimesφfunctionninteger-minusφfunction . \[ \mathopen{}\left\lvert{}\int _{\mathbb{T}}{}g\mathopen{}\left( z\right)\mathclose{}{φ}_{n}\mathopen{}\left( z\right)\mathclose{}\,\mathrm{d}z-\int _{\mathbb{T}}{}g\mathopen{}\left( z\right)\mathclose{}φ\mathopen{}\left( z\right)\mathclose{}\,\mathrm{d}z\right\rvert\mathclose{}\leq g\mathopen{}\left\lVert{}{φ}_{n}-φ\right\rVert_\infty\mathclose{} \text{.} \]

For any arc Aarc\( A \) in T\( \mathbb{T} \), χAarccharacteristic function ofAarc\( \chi_{A} \) is the L2Lebesgue space\( \mathrm{L}^{\mathrm{2}} \)-limit of a sequence of continuous functions on T\( \mathbb{T} \), so integralTgfunction(zcomplex number of modulus one)timesχAarccharacteristic function ofAarc(zcomplex number of modulus one)dzcomplex number of modulus one=equals0zero \( \int _{\mathbb{T}}{}g\mathopen{}\left( z\right)\mathclose{}\chi_{A}\mathopen{}\left( z\right)\mathclose{}\,\mathrm{d}z= 0 \). For every measurable subset Smeasurable subsetsubsetT\( S\subseteq \mathbb{T} \) we get a sequence (sequenceUunion of arcsninteger)sequenceninteger=1oneinfinity \( \mathopen{}\left({U}_{n}\right)\mathclose{}_{n=1}^{\infty} \) of countable unions of open arcs with Aarc\( A \) and the measure of Uunion of arcsnintegerset differenceSmeasurable subset\( {U}_{n}\setminus S \) converges to zero by the regularity of Lebesque measure. So integralUunion of arcsnintegergfunction(zcomplex number of modulus one)dzcomplex number of modulus one=equals0zero \( \int _{{U}_{n}}{}g\mathopen{}\left( z\right)\mathclose{}\,\mathrm{d}z= 0 \) for all ninteger\( n \) implies integralSmeasurable subsetgfunction(zcomplex number of modulus one)dzcomplex number of modulus one=equals0zero \( \int _{S}{}g\mathopen{}\left( z\right)\mathclose{}\,\mathrm{d}z= 0 \). Thus if gfunction\( g \) is zero almost everywhere, then ffunction\( f \) is zero almost everywhere.

We have shown that ffunction=equalssummationninteger=1oneinfinityffunction, esine functionnintegertimesesine functionninteger \( f= \sum_{n=1}^{\infty}{}\mathopen{}\left\langle{}f, {e}_{n}\right\rangle\mathclose{}{e}_{n} \), and thus ffunction2two=equalssummationninteger=1oneinfinity|modulusffunction, esine functionninteger|modulus2two \( {\mathopen{}\left\lVert{}f\right\rVert\mathclose{}}^{2}= \sum_{n=1}^{\infty}{}{\mathopen{}\left\lvert{}\mathopen{}\left\langle{}f, {e}_{n}\right\rangle\mathclose{}\right\rvert\mathclose{}}^{2} \) for all ffunctionelement ofL2Lebesgue space\( f\in \mathrm{L}^{\mathrm{2}} \). For any choice of ffunction\( f \) that is readily integrable against each esine functionninteger\( {e}_{n} \), we get an amusing series summation. For instance, ffunctionequivalent1identity function\( f\equiv \mathbf{1} \) gives ffunction, esine functionninteger=equals2twotimesintegral0zero1onesinsine(nintegertimesπpitimestreal number)dtreal number=equals{cases2twotimes2twoπpitimesninteger, ninteger|restricted toodd; 0zero, ninteger|restricted toeven.} \[ \mathopen{}\left\langle{}f, {e}_{n}\right\rangle\mathclose{}= \sqrt{2}\int _{0}^{1}{}\sin\mathopen{}\left( n\mathrm{\pi}t\right)\mathclose{}\,\mathrm{d}t= \begin{cases}\frac{2\sqrt{2}}{\mathrm{\pi}n}, & n|\text{odd}; \\ 0, & n|\text{even.}\end{cases} \] Then ffunction2two=equalssummationninteger=1oneinfinity|modulusffunction, esine functionninteger|modulus2two \( {\mathopen{}\left\lVert{}f\right\rVert\mathclose{}}^{2}= \sum_{n=1}^{\infty}{}{\mathopen{}\left\lvert{}\mathopen{}\left\langle{}f, {e}_{n}\right\rangle\mathclose{}\right\rvert\mathclose{}}^{2} \), so 1one=equals8eightπpi2twotimessummationninteger|odd1oneninteger2two \[ 1= \frac{8}{{\mathrm{\pi}}^{2}}\sum_{n|\text{odd}}{}\frac{1}{{n}^{2}} \] implies summationninteger|odd1oneninteger2two =equalsπpi2two8eight \[ \sum_{n|\text{odd}}{}\frac{1}{{n}^{2}} = \frac{{\mathrm{\pi}}^{2}}{8} \]and summationninteger=1oneinfinity1oneninteger2two =equals4four3threetimes(πpi2two8eight)=equalsπpi2two6six . \[ \sum_{n=1}^{\infty}{}\frac{1}{{n}^{2}} = \frac{4}{3}\mathopen{}\left(\frac{{\mathrm{\pi}}^{2}}{8}\right)\mathclose{}= \frac{{\mathrm{\pi}}^{2}}{6} \text{.} \]


Previous: Finite Dimensional Spaces back to top Next: Constructions