I am reading a book on Frobenius algebra in which I read all the following definitions:
Given a linear map $f:V\to W$ the dual map is defined by $f^{*}:W^{*}\to V^{*}$( as, $h\to f\circ h$)
Let there be a pairing $\beta:V \otimes W \to \mathbb{k}$ for each fix $w\in W$ define the linear functional $\beta_{w}:V \to \mathbb{k}$ as, $(v \to <v|w>)$. Now define
$\beta_{left}:W \to V^{*}$ given by $w\to \beta_{w}$ similarly define $\beta_{right}:V \to W^{*}$ given by $v\to \beta_{v}$ ( where, $\beta_{v}:W \to \mathbb{k}$ as, $(w \to <v|w>)$. After all these a paragraph in my book says as follows:
Actual question: For finite-dimensional vector spaces, the dualising functor is a (contravariant) equivalence of categories, so in particular it preserves the property of being invertible. So in this case $\beta_{left}$ is injective if and only if $\beta_{right}$ is.
I tried to prove the both the assertion but could not. Say for second assertion I assume $\beta_{left}$ is injective then how to proceed further I wanted to use $\ker(\beta_{left})=e$ but I dont know what is this $e$ as a functional.
Please help me by giving a solution/hint or a reference book that teaches Frobenius algebra in detailed way.