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I am trying to prove that T'' = T (where T'' is the double transpose of T) by showing that the the dual of the dual of a linear finite vector space is isomorphic to the original vector space.

i.e., T: X --> U (A linear mapping)

The transpose of T is defined as the following:

T': U' --> X' (Here U' is the dual of U and X' is the dual of X)

And the double transpose of T is defined as:

T'': X'' --> U'' (Here X'' is the dual of the dual of X and U'' is the dual of the dual of U)

And since X'' is isomorphic to X and U'' is isomorphic to U, (This is the part I still need to prove)

T'' = T

i.e., T: X --> U (A linear mapping)

The transpose of T is defined as the following:

T': U' --> X' (Here U' is the dual of U and X' is the dual of X)

And the double transpose of T is defined as:

T'': X'' --> U'' (Here X'' is the dual of the dual of X and U'' is the dual of the dual of U)

And since X'' is isomorphic to X and U'' is isomorphic to U, (This is the part I still need to prove)

T'' = T

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