16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 8.1.17
Inner Product Spaces
Mathematics — Linear Algebra · conjugation-inner-products
Problem
Let V be a complex vector space, J a conjugation on V, W the set of α in V such that Jα=α, and f an inner product on W. Show that:
(a) There is a unique inner product g on V such that g(α,β)=f(α,β) for all α, β in W.
(b) g(Jα,Jβ)=g(β,α) for all α, β in V.
What does part (a) say about the relation between the standard inner products on R1 and C1, or on Rn and Cn?