I am reading the proof of theorem 2 in chapter 6 Evan PDE.
I have difficulty verifying the following part of the proof, i.e.
3 questions here.
1) The assumptions $u\in H_0^1(U)$ and $u\in L^2(U)=1$ allow us to write $u=\sum_{k=1}^{\infty}d_k w_k$ for $d_k=(u,w_k)_{L^2(U)}.$
2) How to show the series converge in $L^2(U)? $ Well, I know $u=\sum_{k=1}^{\infty}(u,w_k)w_k$, then $\|u\|_{L^2(U)}=\int_U [\sum_{k=1}^{\infty}(u(x),w_k(x))w_k(x)]^2dx$...
3) I have no idea how to show (9).
Appreciate for any helps.
A lot of this comes down to understanding definitions.
$L^2(U)$ is, by definition, the set of all $u$ of the appropriate domain for which $\|u\|_{L^2(U)}^2 < \infty$. Since $u$ is an element of $L^2(U)$, the definition of an orthonormal basis tells us that we may write $$ u = \sum_{k=1}^\infty d_k w_k $$ for an appropriate choice of coefficients $d_k$.
(2) and (3) come from a property of general Hilbert spaces. In particular: if $H$ is a Hilbert space and the elements $\{w_k\}$ are orthonormal elements, then for any coefficients $d_k$, we have $$ \left\|\sum_{k=1}^\infty d_k w_k\right\|_{H}^2 = \sum_{k=1}^\infty |d_k|^2 $$ which of course means that $\sum_{k=1}^\infty |d_k|^2 < \infty$ if and only if $\sum_{k=1}^\infty d_k w_k$ is an element of the Hilbert space. To see why this is true, it suffices to note that the norm on a Hilbert space comes from an inner product, so that $$ \left\|\sum_{k=1}^\infty d_k w_k\right\|_{H}^2 = \left( \sum_{k=1}^\infty d_k w_k, \sum_{\ell=1}^\infty d_\ell w_\ell \right)_H = \sum_{k=1}^\infty\sum_{\ell=1}^\infty d_k \overline{d_\ell} (w_k, w_\ell)_H $$ then, by the definition of an orthonormal basis, we have $$ (w_k, w_\ell)_H = \begin{cases} 1 & k = \ell\\ 0 & k \neq \ell \end{cases} $$