The question that I am stuck on is: Show that if X" (double dual of X) is identified with X and U" (double dual of U) with U via the duality relation, then T" (double transpose) = T. (Duality relation is f(L) = L (x) where f is in X", L is in X', and x is in X) So far, here is my work: We know: T: X --> U is a linear homogeneous map Therefore, T': U' --> X' where U' is the dual of U and X' is the dual of X Then, T": X" --> U" where X" is the double dual of X and U" is the double dual of U. Also, X" is isomorphic to X, and U" is isomorphic to U. I am missing something here, however. This is where I am stuck. How can one deduce that, in fact, T" = T? How do we show that two linear homogenoue maps are the equivalent? The idea of a double dual has left me slightly confused and any help would REALLY be appreciated. Thanks.