Countable Orthonormal Basis for Hilbert Space

Actually, we have a non-constructive proof.Lemma : Let \$mathcalH\$ be a Hilbert space. Then it is impossible

that \$mathcalH\$ has a countable dense subset \$D\$ and an uncountable

orthonormal set \$e_imid iin I\$.Proof: Prove by contradiction. Suppose the contrary that \$mathcalH\$

has a countable dense subset \$D\$ and an uncountable orthonormal set

\$e_imid iin I.\$ Let \$rfrac14\$ and for each \$iin I\$,

let \$B_iB(e_i,r)\$, then open ball centered at \$e_i\$ with

radius \$r\$. Note that \$B_icap B_jemptyset\$ whenever \$ineq j\$.

(For, if there exists \$xin B_icap B_j\$, then \$||e_i-e_j||leq||e_i-x||||x-e_j||e_i\$, where the sum is independent

of ordering. (I leave the proof to you)./////////////////////////////////////////////////////////////I expand the last line, state it as a theorem and give a self-contained proof. This is known as the Fourier expansion in Hilbert space.////////////////////////////////////////////////////////////Theorem: Let \$mathcalH\$ be a Hilbert space (over \$mathbbR\$

or \$mathbbC\$) and let \$e_imid iin I\$ be a maximal (with

respect to \$subseteq\$) orthonormal set (where the index \$I\$ may

be uncountable). Then for each \$xinmathcalH\$, we have \$xsum_iin Ilangle x,e_irangle e_i\$,

where the series converges in unorder sense (explained in Claim (4)

below).Proof: Claim (1) (Bessel's inequality): For any finite subset \$I_1subseteq I\$

and any \$xinmathcalH\$, \$sum_iin I_1|langle x,e_irangle|^2leq||x||^2\$. Proof of Claim (1): Denote \$Pxsum_iin I_1langle x,e_irangle e_i\$.

Observe that \$Pxbot x-Px\$, so \$||x||^2||Px||^2||x-Px||^2geq||Px||^2sum_iin I_1|langle x,e_irangle|^2\$.Claim (2): For any \$xinmathcalH\$, \$iin Imidlangle x,e_irangleneq0\$

is a countable set.Proof of Claim (2): Let \$I'iin Imidlangle x,e_irangleneq0\$.

Prove by contradiction. Suppose the contrary that \$I'\$ is uncountable.

For each \$ninmathbbN\$, let \$I_niin Imid|langle x,e_irangle|geqfrac1n\$.

Observe that \$I'cup_nI_n\$, so there exists \$n\$ such that \$I_n\$

is uncountable. Choose \$k\$ such that \$frackn^2>||x||^2\$.

Choose a finite subset \$I'_nsubseteq I_n\$ that contains \$k\$

elements, then \$sum_iin I'_n|langle x,e_irangle|^2geqfrackn^2>||x||^2\$,

contradicting to claim (1).Claim (3): Given \$xinmathcalH\$ and let \$I_xiin Imidlangle x,e_irangleneq0\$.

Fix an enumeration for \$I_x\$, say \$I_xi_1,i_2,ldots\$

(finite or infinite), then the series \$sum_klangle x,e_i_krangle e_i_k\$

is convergent.Proof of Claim (3): If \$I_x\$ is a finite set, we are done. Suppose

that \$I_x\$ is a countably infinite set. Let \$s_nsum_k1^nlangle x,e_i_krangle e_i_k\$.

By Claim (1), for each \$n\$, \$sum_k1^n|langle x,e_i_krangle|^2leq||x||^2\$,

so the series \$sum_k1^infty|langle x,e_i_krangle|^2\$

is convergent. We show that \$s_n\$ is a Cauchy sequence in \$mathcalH\$.

Let \$varepsilon>0\$. Choose \$N\$ such that \$sum_kN^infty|langle x,e_i_krangle|^20\$

be arbitrary. Let \$U_yB(y,varepsilon)\$ be the open ball centered

at \$y\$ with radius \$varepsilon\$. Choose \$N\$ such that \$||sum_k1^Nlangle x,e_i_krangle e_i_k-y||leqfracvarepsilon4\$.

(If \$I_x\$ in Claim (3) is a finite set, let \$N\$ be the number

of elements in \$I_x\$.) We adopt the notation in Claim (3) and continue

to work with the enumeration \$I_xi_1,i_2,ldots\$. Let

\$I_1i_1,i_2,ldots,i_N\$. Clearly \$I_1inmathcalC\$.

Consider the case that \$I_x\$ is an infinite set (the finite case

is trivial). By continuity of norm (i.e., the continuity of the map

\$xmapsto||x||\$), we have

\$\$

lim_n||sum_k1^Nlangle x,e_i_krangle e_i_k-sum_k1^nlangle x,e_i_krangle e_i_k||||sum_k1^Nlangle x,e_i_krangle e_i_k-y||leqfracvarepsilon4.

\$\$

On the other hand, for any \$n>N\$, we have \$||sum_k1^Nlangle x,e_i_krangle e_i_k-sum_k1^nlangle x,e_i_krangle e_i_k||^2sum_kN1^n|langle x,e_i_krangle|^2\$.

Hence \$sum_kN1^infty|langle x,e_i_krangle|^2leqleft(fracvarepsilon4right)^2\$.

Let \$I_2inmathcalC\$ be arbitrary such that \$I_1preceq I_2\$.

Then

\$\$

||theta(I_2)-y||leq||sum_k1^Nlangle x,e_i_krangle e_i_k-y||||sum_iin I_2setminus I_1langle x,e_irangle e_i||leqfracvarepsilon4||sum_iin I_2setminus I_1langle x,e_irangle e_i||.

\$\$

Observe that

\$\$

||sum_iin I_2setminus I_1langle x,e_irangle e_i||^2sum_iin I_2setminus I_1|langle x,e_irangle|^2leqsum_kN1^infty|langle x,e_i_krangle|^2leqleft(fracvarepsilon4right)^2

\$\$ because for any \$iin I_2setminus I_1\$, if \$inotini_N1,i_N2,ldots\$,

then \$langle x,e_irangle0\$.Now we have: \$||theta(I_2)-y||leqfracvarepsilon2\$. This

shows that \$sum_iin Ilangle x,e_irangle e_iy\$ in unordered

sense.Claim (5): The \$y\$ defined in Claim (3) and Claim (4) is \$x\$. That

is \$xsum_iin Ilangle x,e_irangle e_i\$.Proof of Claim (5): Let \$zx-y\$. Prove by contradiction. Suppose

the contrary that \$zneq0\$. We adopt the notation in Claim (3) and

Claim (4). Recall that for each \$ainmathcalH\$, the map \$xmapstolangle x,arangle\$

is continuous. Let \$iin I\$ be arbitrary. Conside the case that \$I_x\$

is infinite (The finite case is trivial.). We have

\$\$

langle z,e_iranglelim_nrightarrowinftylangle x-sum_k1^nlangle x,e_i_krangle e_i_k,e_irangle.

\$\$ If \$inotin I_x\$, we have \$langle x,e_irangle0\$ and \$langle e_i_k,e_irangle0\$

for all \$k\$, and hence \$langle z,e_irangle0\$. Suppose that

\$iin I_0\$, say \$ii_k'\$ for some \$k'inmathbbN\$. Then for

any \$ngeq k'\$, we have

\$\$

langle x-sum_k1^nlangle x,e_i_krangle e_i_k,e_iranglelangle x,e_irangle-langlesum_k1^nlangle x,e_i_krangle e_i_k,e_iranglelangle x,e_i_k'rangle-langle x,e_i_k'rangle0.

\$\$ Therefore \$langle z,e_irangle0\$ in all cases. Define \$tildezz/||z||\$,

then \$e_imid iin Icuptildez\$ is an orthonormal set,

containing \$e_imid iin I\$ as a proper subset. This contradicts

to the maximality of \$e_imid iin I\$.

If \$Dx_n : nin mathbbN\$ is a countable set dense in a Hilbert space \$mathcalH\$, how can I show that Gram-Schmidt algorithm applied to \$D\$ (or a subset of \$D\$) produces an orthonormal numerable basis for \$mathcalH\$?

So far I have been able to prove that every ortonormal basis of \$mathcalH\$ has to be numerable.

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