Operators on Complex Vector Spaces — Sheldon Axler
Mathematics — Linear Algebra · operator-norms-functionals
Problem
Prove that if u∈V and φu is the linear functional on V defined by the equation φu(v)=⟨v,u⟩, then ∥φu∥=∥u∥.
Here we are thinking of the scalar field F as an inner product space with ⟨α,β⟩=αβˉ for all α,β∈F. Thus ∥φu∥ means the norm of φu as a linear map from V to F.