Let \(H=L^2(\mathbb R)\) be the Hilbert space with the inner product \(\int _{\mathbb R}f(x)\bar g(x)\,dx\), where \(dx\) is the Lebesgue measure. Let \(T:H\to M\) be the orthogonal projection of \(H\) onto a finite dimensional subspace \(M\) of \(H\).
Show that there exists a function \(h:\mathbb R^2\to \mathbb C\) such that \[ Tf(x)=\int _{\mathbb R}h(x,y)f(y)\,dy \] for all \(f\in L^2(\mathbb R)\).
(Hint: Let \(u_1,\ldots ,u_n\) be an orthonormal basis of \(M\). You may express \(h\) in terms of \(u_1,\ldots ,u_n\).)