Bilinear form/Linear forms/Nondegenerate/Fact

Let ${}K$ be a field, and let ${}V$ denote a ${}K$-vector space, endowed with a bilinear form ${}\left\langle -,-\right\rangle$. Then the following statements hold.

1. For every vector ${}u\in V$, the assignments

   $V\longrightarrow K,v\longmapsto \left\langle u,v\right\rangle ,$

   and

   $V\longrightarrow K,v\longmapsto \left\langle v,u\right\rangle ,$

   are ${}K$-linear.

2. The assignment

   $V\longrightarrow {V}^{*},u\longmapsto \left\langle u,-\right\rangle ,$

   is ${}K$-linear.

3. If ${}\left\langle -,-\right\rangle$ is nongenerate, then the assignment in (2) is injective. If, moreover, ${}V$ is finite-dimensional, then this assignment is bijective.

Proof, Write another proof