- #1

nyisles131

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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.