Question

Let P∞ be the vector space consisting of all polynomial functions. Which of the following are subspaces of P∞? Justify your answer in each case.
(a) The set of polynomials f(x) satisfying f(1) = 0.
(b) The set of polynomials f(x) satisfying f(0) = 1.
(c) The set of polynomials f(x) satisfying f(0) = f '(1).
(d) The set of polynomials f(x) satisfying f(0)f(1) =0.
(e) The set of polynomials of the form ax
2 + bx + a, where a, b are any real numbers.