(a) Show that C[a, b] is a real vector space. (b) Let W (k) be the set...

Question:

(a) Show that C[a, b] is a real vector space.
(b) Let W (k) be the set of all functions in C[a, b] with ((a) = k. For what values of k will W (k) be a sub-space of C[a, b]?
(c) Let t1, t2, ( ( ( ( tn be a fixed set of points in [a, b]. Show that the subset of all functions ( in C[a, b] that have roots at t1, t2, ( ( ( ( tn, that is, ((ti) = 0 for i = 1,2, ( ( ( ( n, forms a subspace.