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I am reading lecture note on real analysis and found following two assertions that I think are true but unable to give formal proofs.

Let $\{\varphi_n\}$ be a orthonormal basis of $L^2(E)$ and $f\in L^2(E)$, then $\lim_{n\rightarrow\infty}\left<f,\varphi_n\right>=0.$

Let $\{\varphi_n\}$ be a orthonormal basis of $L^2(A)$ and $\{\phi_n\}$ be a orthonormal basis of $L^2(B).$ Then $\{\varphi_n(x)\cdot \phi_m(y)\}$ is an orthonormal basis of $L^2(A\times B).$

I am very sorry if this question is very basic and thanks for your help in advance.

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  • Your first concern is a very important point, but is not something really about $L^2$ or functions. Let $H$ be a Hilbert space and ${e_n}{n\in I}$ an ONB of $H$. This means that each $v\in H$ has a unique decomposition $v=\sum{n\in I} v_n , e_n$. Can you calculate what $\langle v, e_k\rangle$ is in terms of the $v_n$? What about an expression for $|v|^2$ involving the $v_n$? Make use of the fact the $e_n$ are all mutually orthogonal and of norm $1$. – s.harp Nov 11 '19 at 14:04

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For your first point you may notice that, as $\{\varphi_{n}\}$ is an orthonormal basis, for every $f \in L^{2}(E)$ you have $$ f = \sum_{n=1}^{\infty}{\langle f, \varphi_{n} \rangle \varphi_{n}}$$ with $\|f\|^{2} = \sum_{n=1}^{\infty}{|\langle f, \varphi_{n} \rangle|^{2}}$, so $|\langle f, \varphi_{n} \rangle|^{2} \to 0$ as $n \to \infty$, which yields to your first assertion.

For the second one, this question may help you Orthonormal basis for product $L^2$ space

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