Let $X$ be a vector space over an arbitrary field $\mathbb{F}$ and denotes its dual by $X^*$. Suppose $k:X\times X^*\to\mathbb{F}$ be a bilinear map.
How can I prove that there exists a linear map $f:X\to X$ such that $k(x,x^*) = x^*(f (x))$ for every $x\in X$ and all $x^*\in X^*$.