Bilinear form/Via mapping to dual space/Fact/Proof

Proof

Since ${}\Theta (u)\in {V}^{*}$, the evaluation at a vector ${}v\in V$ yields an element of the base field. The linearity in the second component rests on the fact that ${}\Theta (u)$ belongs to the dual space. The linearity in the first component rest on the linearity of ${}\Theta$.

To fact