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Let $(X,\mu)$ and $(Y,\nu)$ be $\sigma$-finite measure spaces such that $L^2(X)$ and $L^2(Y)$ . Let $\{f_n\}$ be an orthonormal basis for $L^2(X)$ and let $\{g_m\}$ be an orthonormal basis for $L^2(Y)$. I am trying to show that $\{f_n g_m\}$ is an orthonormal basis for $L^2(X\times Y)$. So far, I have attempted to show that if $h\in L^2(X\times Y)$, then

\begin{align} h(x,y) &= \sum_m \sum_n \langle h , f_n g_m\rangle\,f_mg_m \\[0.3cm] &= \sum_m \sum_n \left(\int_X \int_Y h(r,s) f_n(r)g_m(s) d\nu(s) d\mu(r)\right) f_n(x)g_m(y).\end{align}

Using the fact that $x\mapsto h(x,y) \in L^2(X)$ for almost every $y$ and similarly for $y\mapsto h(x,y)$, I can obtain

$h(x,y) = \sum_{m=1} ^\infty (\int_X [\sum_{n=1} ^\infty (\int_Y h(r,s)g_n(s) d\nu(s)) f_m(r)] d\mu(r)) f_m(x)g_n(y)$.

However, I am unable to justify passing the summation outside the integral. Any suggestions?

Martin Argerami
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user15464
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1 Answers1

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The way I would do it is first to show that $\{f_n\times g_m\}$ is orthonormal: indeed, $$ \langle f_n\times g_m, f_s\times g_t\rangle = \langle f_n,f_s\rangle\,\langle g_m,g_t\rangle =\delta_{n,s}\,\delta_{m,t} = \delta_{(n,m),(s,t)}. $$

And then it only remains to show that it is a basis; for that, we can show that its orthogonal complement is zero. So, if $\langle h, f_n\times g_m\rangle = 0 $ for all $n,m$, we have $$ 0=\int_X\left(\int_Y h(x,y)\,g_m(y)\,d\nu(y)\right)\,f_n(x)\,d\mu(x); $$ so, as $n$ is arbitrary, the function $x\mapsto\int_Y h(x,y)\,g_m(y)\,d\nu(y)$ is zero almost everywhere for each $m$. Let $$ E_m=\{x\in X: \int_Y h(x,y)\,g_m(y)\,d\nu(y)\ne0\}. $$ Each $E_m$ is a null-set, and then so is its (countable) union $E$. Outside of $E$, $$ \int_Y h(x,y)\,g_m(y)\,d\nu(y)=0\ \ \ \mbox{for all } m. $$ Thus for each $x\in X\setminus E$, $h(x,y)=0$ almost everywhere. As $|h|^2$ is integrable, its integral agrees with the iterated integrals, so $$ \int_{X\times Y} |h(x,y)|^2\, d(\mu\times\nu)=\int_X\int_Y|h(x,y)|^2\,d\nu(y)\,d\mu(x) =\int_{X\setminus E}\int_Y|h(x,y)|^2\,d\nu(y)\,d\mu(x)=0. $$ So $h=0$ in $L^2(X\times Y)$.

Martin Argerami
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  • Why is it important that the measure spaces be $\sigma$-finite? – RitterSport Jul 07 '17 at 20:23
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    Because otherwise the bases are not countable, which ruins the argument with the null set. – Martin Argerami Jul 07 '17 at 22:54
  • @MartinArgerami: You are tacitly assuming that the map $x \mapsto \int_Yh(x,y)g_m(y) , d\nu(y)$ is in $L^2(X)$, so that taking the inner product with the $f_n(x)$ shows that this map vanishes. How do you know that it is indeed in $L^2(X)$? – Adrian Clough Jan 06 '18 at 17:04
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    It follows directly from $h\in L^2$: \begin{align} \int_X\left|\int_Y h(x,y),g_m(y),d\nu(y)\right|^2,d\mu(x)&\leq\int_X,\int_Y|h(x,y)|^2,d\nu(y),\int_Y|g_m(y)|^2,d\nu(y),d\mu(x)\ \ \ &=\int_X,\int_Y|h(x,y)|^2,d\nu(y),d\mu(x)=|h|^2.\end{align} – Martin Argerami Jan 06 '18 at 18:01
  • Ah, thank you for your clarification! – Adrian Clough Jan 06 '18 at 18:27
  • Why can you write $\langle f_n\times g_m, f_s\times g_t\rangle = \langle f_n,f_s\rangle,\langle g_m,g_t\rangle$? More specifically, why is Fubini's theorem usable for the product of two $L^2$ functions here? Thanks. – Longti Feb 22 '18 at 20:37
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    It's not really Fubini. Basically what I'm saying is that $$\int_X\int_Yf(x)g(y),dy,dx=\int_Xf(x)\int_Yg(y),dy,dx=\left(\int_Yg(y),dy\right),\int_Xf(x),dx.$$If you want, technically you need to justify that the integral is the double integral, but that follows from the definition of product measure when your function is separable (i.e., $f(x)g(y)$). – Martin Argerami Feb 22 '18 at 20:55
  • @MartinArgerami Thank you for the help! – Longti Feb 23 '18 at 03:44
  • @Martin Argerami How did you put the square of the entire integeal into the square of just the integrand? Because there is no limit on the measure of $Y$, the Jensen formula does not seem to hold... – Keith May 01 '19 at 21:01
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    If you mean my comment from Jan 6 at 18:01, it is simply Cauchy-Schwarz (or Hölder, if you prefer). – Martin Argerami May 01 '19 at 21:33
  • Oh.. I missed the simple and clear inequalities.... – Keith May 01 '19 at 21:41