Problem 5. (i) Let U be a finite-dimensional subspace of V. Prove that Ut = {0} if and only if U = V. (ii) Let C([-1,1])

[answerhappygod]

Problem 5 I Let U Be A Finite Dimensional Subspace Of V Prove That Ut 0 If And Only If U V Ii Let C 1 1 1 (60.53 KiB) Viewed 49 times

Problem 5. (i) Let U be a finite-dimensional subspace of V. Prove that Ut = {0} if and only if U = V. (ii) Let C([-1,1]) be the vector space of continuous real-valued functions on the interval [-1, 1] with inner product given by (5,9) = 5 f(x)g(x)dx for f, 9 € CC[-1,1). Let U be the subspace defined by U = {f e C([-1,1]): f(0) = 0}. Show that U+ = {0}. -1 Note: The second question tells us that the assumption U is finite-dimensional is necessary for the first question.