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Exercises 5.8 Exercises

1. Orthogonal Complements (level 2).

Let \(M\) be a subspace of a Hilbert space. Show that \((M^\perp)^\perp = M\) if and only if \(M\) is closed.

2. Projections (level 2).

Let \(H\) be a Hilbert space with orthonormal basis (ONB) \(\{\psi_i\}_{i\geq 1} \text{.}\) For each \(n\geq 1\) define the projection operator \(P_n\) as
\begin{equation*} P_n u =\sum_{i=1}^n (\psi_i, u) \psi_i . \end{equation*}
  1. Show that \(P_n\) is a projection, i.e. \(P_n^2=P_n\text{.}\)
  2. Show that \(P_n\) is self-adjoint.
  3. The orthogonal complement of \(P_n\) is \(Q_n=\mathbb I - P_n\text{.}\) Show that for each \(u\text{:}\)
    \begin{equation*} \lim_{n\to \infty} \|Q_n u \| =0. \end{equation*}

3. Dense Subspaces (level 3).

Let \(V\) be a linear subspace of a Banach space \(X\text{.}\) Show that \(V\) is not dense in \(X\) if and only if there exists \(l\in X^*\text{,}\) \(l\neq 0\) such that \(l(x) = 0 \) for all \(x\in V\text{.}\)

4. weak convergence (level 1).

Let \(T\in {\cal L}(X,Y)\) and \(X,Y\) are Banach spaces. Let \(x_n\) be a weak convergent sequence in \(X\text{.}\) Show that \(Tx_n\) converges weakly in \(Y\text{.}\)

5. strong convergence (level 1).

Let \(H\) be a Hilbert space. Let \(x_n\) converge weakly to \(x\) and assume in addition that the norms converge as well: \(\|x_n\|\to \|x\|\text{.}\) Show that \(x_n\to x\) strongly in \(H\text{.}\)