Ubungen zu Analyis II¨ Blatt 12
1 Prove thatL(E1, . . . , En;F) is a Banach space, ifF is complete.
2 Prove that the isomorphism betweenL(E, Lk−1(E, F)) andLk(E, F) is norm preserving for allk≥2.
3 LetE be a complex normed space andϕ∈E∗. Prove the identity ϕ(x) = Reϕ(x)−Reϕ(ix) ∀x∈E
4 Consider in the real Hilbert spacel2the basis elementsei thei-th components of which are 1 while all other components are 0 and prove
ei+0, but theei do not converge in the norm topology.