91Ó°ÊÓ

What are the adjoints of a zero operator 0 and an identity operator \(I ?\)

Show that a Banach space \(X\) is reflexive if and only if its dual space \(X\) ' is reflexive. (Hint. It can be shown that a closed subspace of a reflexive Banach space is reflexive. Use this fact, without proving it.)

Let \(X\) and \(Y\) be normed spaces and \(X\) compact. If \(T: X \longrightarrow Y\) is a bijective closed linear operator, show that \(T^{-1}\) is bounded.

(Null space) Show that the null space \(\mathcal{N}(T)\) of a closed linear operator \(T: X \longrightarrow Y\) is a closed subspace of \(X\).