The double dual space of V, denoted V 00, is defined to be the dual spaceof V 0. In
Chapter 3, Problem 34(choose chapter or problem)
The double dual space of V, denoted V 00, is defined to be the dual spaceof V 0. In other words, V 00 D .V 0/0. Define W V ! V 00 by.v/.'/ D '.v/for v 2 V and ' 2 V 0.(a) Show that is a linear map from V to V 00.(b) Show that if T 2 L.V /, then T 00 D T, where T 00 D .T 0/0.(c) Show that if V is finite-dimensional, then is an isomorphismfrom V onto V 00.[Suppose V is finite-dimensional. Then V and V 0 are isomorphic, butfinding an isomorphism from V onto V 0 generally requires choosing abasis of V. In contrast, the isomorphism from V onto V 00 does notrequire a choice of basis and thus is considered more natural.]
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