Separability, Bessel's Inequality, and Orthonormal Bases

Show that an orthonormal subset of a separable Hilbert space $H$ must be countable.

Let $\left\{ \varphi_{k}\right\}$ be an orthonormal sequence in a Hilbert space $H$. Show that $\left\{ \varphi_{k}\right\}$ converges weakly to 0 in $H$.

Let $\left\{ \varphi_{k}\right\}$ be an orthonormal basis for the separable Hilbert space $H$. Show that $\left\{ u_{n}\right\} \rightarrow u$ weakly if and only if $\left\{ u_{n}\right\}$ is bounded and, for each $k, \lim_{n \rightarrow \infty } \left\langle u_{n}, \varphi_{k} \right\rangle = \left\langle u, \varphi_{k} \right\rangle$.

Show that any two infinite dimensional separable Hilbert spaces are isometrically isomorphic and that any such isomorphism preserves the inner-product.

Let $V$ a closed, separable subspace of $H$ for which $\left\{ \varphi_{k}\right\}$ is an orthonormal basis. Show that the orthogonal projection of $H$ onto $V, P$, is given by

Let $\left\{ \varphi_{k}\right\}$ be an orthonormal basis for a Hilbert space $H$. Verify that

Also verify that

where, for each natural number $k, a_{k} = \left\langle u, \varphi_{k} \right\rangle$ and $b_{k} = \left\langle v, \varphi_{k} \right\rangle$.

Verify the assertions in the example of the orthonormal basis for $L^{2}[0,2 \pi ]$.

Use Proposition 5 and the Stone-Weierstrass Theorem to show that for each $f \in L^{2}[-\pi , \pi ]$,

where the convergence is in $L^{2}[-\pi , \pi ]$ and each

Proposition 5: An orthonormal sequence $\left\{ \varphi_{k}\right\}$ in a Hilbert space $H$ is complete if and only if it is an orthonormal basis.