Problem 6. Let V be a vector space. (a) Let (-, -) : V x V - R be an inner product. Prove that (-, -) is a bilinear form on V. (b) Let 3 = (U1, . .., Un) be a basis of V. Prove that there exists a unique inner product on V making orthonormal. (c) Let (V) be the set of all inner products on V. By part (a), F(V) C B(V). Is $ (V) a vector subspace of B(V)? (d) Suppose that V is finite-dimensional, and let S be a set of inner products on V that is (when viewed as a subset of B(V)) linearly independent. Prove that S must be finite. (e) Consider the vector space V = R[X] of all polynomials with real coefficients. For each n E N, define the function (-, -)n : V x V -> R by (do + alX + . . . + amx", bo + b1X + . ..+ bmx" ) = 2aobo + (albi + . . . + ambm) + dobr + anbo. prove that (-, -), is an inner product for all n E N. (f) Using the previous part, exhibit an infinite linearly independent set of inner products on [X]