(Vector Space of Random Variables) To draw a parallel between the Euclidean vector space E^J and a vector space of random variables, consider a discrete random variable y with J possible outcomes and the set of random variables f(y) that can be generated as real functions f(.) of y.

(a) Show that the set of random variables {f(y) | f: R → R^J} is a vector space.

(b) Show that this vector space has a dimension equal to J.

(c) Show that the expectation E[f₁(y)f₂(y)] is an inner product on this vector space. Give a matrix representation of this inner product.

(d) Describe the norm corresponding to this inner product.