Exercise 2, § 4

Let E and F be Banach spaces and G a subspace of E. Let $u$ be a linear mapping of finite rank from G into F. Suppose that $u$ is not continuous on G.

a) The partial operator from E into F with domain G defined by $u$ is not closable.

b) Suppose that E and G are Hilbert spaces. Calculate the adjoint of the partial operator defined by $u$.