Let \((X,\mathfrak M,\mu )\) be a positive measure space such that \(\mu (X)=1\). If \(V\) is a closed subspace of \(L^2(\mu )\) which is contained in \(L^\infty (\mu )\) then show that \(\dim (V)<\infty \) as follows:
(1) Show that there exists \(C>0\) such that \(\lVert f\rVert _\infty \le C\lVert f\rVert _2\) for all \(f\in V\).
(2) Let \(\{f_1,\ldots ,f_n\}\) be any orthonormal set in \(V\) and show that for all \(c_1,\ldots ,c_n\in \mathbb C\), and almost all \(x\in X\), \[ \left |\sum _{i=1}^{n}c_i f_i(x)\right | \le C\left (\sum _{i=1}^{n}|c_i|^2\right )^{1/2}. \]
(3) Choose \(c_i=\overline {f_i(x)}\) in (2) and deduce that \[ \sum _{i=1}^{n}|f_i(x)|^2\le C^2\qquad \text {(for almost all }x\in X\text {)}. \]
(4) Integrate the inequality in (3).