3. A finite-dimensional Hilbert space is a finite-dimensional complex vector space H together with an inner product (,). This means that (x, y) = C for all x, y H and (x, x) 0 implies x = 0; (x, x) > 0 for all x H; (x, y) = (y, x); for any a C, (ax+y, w) a (x, w) + (y, w). = (a) Show that (x, ay+w) = (x, y) + (x, w) for all x, y, w H, a C. (b) Show that C" with (x, y) =1,, is a Hilbert space. (c) Show that C" with (x, y) = kay is a Hilbert space. (d) Show that H = {fe C[x] deg f 5) with (f,g) - ff(t) g(t) dt is a Hilbert space. = (e) Show that H = {fe C[x] deg f5} with (f.9) -o f(k)g(k) is a Hilbert space.