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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 $x^*\in X^*$.

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    Maybe working with a Hamel basis of $X$?2012-11-01
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    @azimut Maybe useful link: [How much bumping is too much?](http://meta.math.stackexchange.com/questions/5068/how-much-bumping-is-too-much) at meta.2013-03-07
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    @MartinSleziak: Thank you. I just wasn't aware that a simple retag bumps the question up to the first page.2013-03-07

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