Exercise 14, § 1

a) Let $E$ and $F$ be Hilbert spaces. A continuous linear mapping $u$ from E into F is compact if and only if the image of $u$ contains no closed subspace of infinite dimension.

b) The characterization of a) of compact endomorphisms of E into F does not hold for every pair of Banach spaces $(E,F)$. (One may use Exercise 5, p. 121).