Let be an infinite dimensional vector space over a field , say with basis . Prove that the dual space has strictly larger dimension than does .
We claim that . To prove this, for each let be a copy of . Now define by . By the universal property of direct products, there exists a unique -linear transformation such that for all . We claim that is an isomorphism. To see surjectivity, let . Now define by letting and extending linearly; certianly . To see injectivity, suppose . Then , so that , and thus for all . Thus for all . Since is a basis of , we have . Thus is an isomorphism, and we have .
By this previous exercise, has strictly larger dimension than does .