Measure, Integration & Real Analysis · Problem 6E.16
L^p Spaces — Sheldon Axler
Mathematics — Measure Theory · new-norm-continuity-functional
Problem
Suppose V is a Banach space with norm ∥⋅∥ and that φ:V→F is a linear functional. Define another norm ∥⋅∥φ on V by
∥f∥φ=∥f∥+∣φ(f)∣.
Prove that if V is a Banach space with the norm ∥⋅∥φ, then φ is a continuous linear functional on V (with the original norm).