試題詳解

18. 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(﹒) \in V. Which of the following statements are true?

(A) Both  are subspaces of V.

(B)

(C)

(D) dim V =

(E) Denote dist(f, ) as the distance induced from the defined inner product between any f(﹒) ∈ V and . Then dist((x + 1)², ) = √8/3.