In Pn(R), the vector space (over the reals) of degree n polynomials,
∈ = {p(x) ∈ Pn(R) j p( - x) = p(x) for all x}
and
O = {p(x) ∈ Pn(R) | {(- x) = - p(x) for all x}
are the even and the odd polynomials; p(x) = x2 is even while p(x) = x3 is odd. Show that they are subspaces. Are they complementary? Are they invariant under the differentiation transformation?