Mathematics — Measure Theory · hilbert-spaces-linear-functionals
Problem
(a) Suppose φ and ψ are bounded linear functionals on a Hilbert space V such that ∥φ+ψ∥=∥φ∥+∥ψ∥. Prove that one of φ,ψ is a scalar multiple of the other.
(b) Give an example to show that (a) can fail if the hypothesis that V is a Hilbert space is replaced by the hypothesis that V is a Banach space.