Let U be a closed subspace of L^2(0,1) that is also contained in C([0,1]).

a) Prove there is an M > 0 such that ||f||_sup <= M||f||_L2 for every f in U.

b) Prove that for each 0 <= x <= 1 there is a g_x in U such that

f(x) = <f,g_x> for all f in U and ||g_x||_L2 \leq M.

It is recommended to use the closed graph theorem. My thinking is that C([0,1]) \subset L^2(0,1) are both Banach spaces, U is a Banach space and the identity map on U from C([0,1]) to L^2(0,1) is closed, linear. Then we get that it is bounded by the closed graph theorem and thus part a) is true.

b) I am having a hard time convincing myself this is true. For example g_0 would need to be a continuous function with the property that f(0) = <f,g_0> for every f in U.