試題詳解

12. Let V = C[-1, 1] be the vector space of all continuous functions defined on [-1, 1], and let and denote, respectively, the set of all even and all odd functions in V. Moreover, define an inner product (f, g) := f(x)g(x)dx for any f(⋅), g(⋅) ∈ V. Which of the following statements are true?
(A) Both are subspaces of V.
(B)
(C).
(D) dimV =
(E) Denote dist(f, ) as the distance induced from the defined inner product between any f(⋅) ∈ and . Then dist((x+1)², ) = √8/3.