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This seems simple, but I just can't quite convince myself. I'm sure someone out there can help.

Let U be a finite dimensional vector space with dual U$^*$ and let B = {$f_1,f_2,...f_n$} be a basis for U$^*$. Let V and W be subspaces of U such that V $\bigcup$ W = U, with duals V$^*$ and W$^*$. Questions:

  1. Is it true that the duals of V and W are the restrictions of some elements of B to V and W respectively?
  2. Is it true that $U^* = V^+ \bigcup W^+$ where the + means that the basis elements of V$^*$ and W$^*$. have been re-extended into their counterparts in B?
  3. If 2 is not true, is there any restriction on V and W which would make it true, for example that $V \bigcap W$ = 0?
  4. If any of the answers above are "yes", do they hold also for spaces of infinite dimension? And if not in general, do they hold at least for reflexive spaces?
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