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.

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

is ${}K$ -linear.
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