I was attempting to solve part $b$ of the following problem from Michael Reed's Functional Analysis book. I got stuck a little bit. I guess in here a "basis" means what it means in linear algebra so $\{\phi_k\}$ is not necessarily am orthogonal basis.
Note that in this book $(\gamma.a,\lambda.b) = \bar{\gamma} . \lambda . (a,b) $
Here's what I've got:
$\begin{align} g(x) = \sum_{k=1}^{\infty}{(\phi_k, g(x)).\phi_k} &=\sum_{k=1}^{\infty}{((\sum_{l = 1}^{\infty}{(e_l, \phi_k). e_l)}, g(x))).(\sum_{l = 1}^{\infty}{(e_l, \phi_k). e_l)}} \\ &= \sum_{k=1}^{\infty}{(\sum_{l = 1}^{\infty}{(\phi_k, e_l)(e_l, g(x))}).(\sum_{l = 1}^{\infty}{(e_l, \phi_k). e_l)}} \end{align}$
but got nothing more. Should I really use the fact that $g$ belongs to $L^2(X,\mu,\mathbb{H})$?
If so, I guess I have to make some orthogonal basis for $\mathbb{H}$ with the aid of $\{\phi_k\}$s, and again if so, how should I?
Are $\{\phi_k\}$s orthogonal in the first place?
Thank you for your attention.
