Measure, Integration & Real Analysis · Problem 6D.1
L^p Spaces — Sheldon Axler
Mathematics — Measure Theory · linear-functionals-bounded
Problem
Suppose V is a normed vector space and φ is a linear functional on V. Suppose α∈F∖{0}. Prove that the following are equivalent.
(a) φ is a bounded linear functional.
(b) φ−1(α) is a closed subset of V.
(c) φ−1(α)=V.