Bilinear form/Via mapping to dual space/Fact

Let ${}K$ be a field, let ${}V$ denote a ${}K$-vector space with its dual space ${}{V}^{*}$. Let

$\Theta \colon V\longrightarrow {V}^{*}$

be a linear mapping.

Then

${}\Psi (u,v)=(\Theta (u))(v)\,$

defines a bilinear form

on ${}V$.

Proof, Write another proof