The dual basis of an infinite dimensional vector space does not span the dual space

Let F be a field and let V be an infinite dimensional vector space over F; say V has basis B = \{v_i\}_I. Prove that the dual basis \{\widehat{v}_i\}_I does not span the dual space \widehat{V} = \mathsf{Hom}_F(V,F).


Define a linear transformation \varphi on V by taking v_i to 1 for all i \in I. Note that for all i \in I, \varphi(v_i) \neq 0. Suppose now that \varphi = \sum_{i \in K} \alpha_i\widehat{v}_i where K \subseteq I is finite; for any j \notin K, we have (\sum \alpha_i \widehat{v}_i)(v_j) = 0, while \varphi(v_j) = 1. Thus \varphi is not in \mathsf{span}\ \{\widehat{v}_i\}_I.

So the dual basis does not span \widehat{V}.

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