a. Let V = C1(I) be the vector space of continuously differentiable functions on the

a. Let V = C1(I) be the vector space of continuously differentiable functions on the open interval I = (0, 1). Define T : V V by T (f )(t) = tf _ (t). Prove that every real number is an eigenvalue of T and find the corresponding eigenvectors. b. Let V = {f C0(R) : limt f (t)|t |n = 0 for all positive integers n}. (Why is V a vector space?) Define T : V V by T (f )(t) = _ t f (s)ds. (If f V , why is T (f) V ?) Find the eigenvalues and eigenvectors of T .