Mathematics · Vector Spaces and Inner Products

Let C[0, π] be the real vector space of real-valued continuous functions on the closed interval [0, π]. For positive integers n, define fn ∈ C[0, π] by: fn(x) = sin(nx)/sin(x) if x ∈ (0, π); n if x = 0; and (-1)n-1n if x = π. Let V be the real subspace of C[0, π] spanned by {f1, f2, f3}. Consider the inner product on V given by: ⟨f, g⟩ = 2π ∫0π f(x)g(x) sin2(x) dx. Which of the following statements is true?