- #1
nyisles131
- 7
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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.
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.