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I am trying to do the next exercise (Exercise 11, section 10.2, Hoffman & Kunze).

Exercise 11, section 10.2, Hoffman & Kunze

I defined a linear functional $L: V \to V^*$ as $(L_f\alpha)(\beta)=f(\alpha, \beta)$. It is easy to show that L is bijective. Then, for a $\beta \in V$ fixed and $T \in \mathcal L(V)$, I defined $$g(\alpha):= f(T\alpha,\beta)$$

So, $g \in V^*$.

With that in mind, I showed the existence of $T$. I could do the rest of the exercise without any problem. However, I think I did not use properly the hypothesis that $f$ is symmetric and, as an additional exercise, I want to show the same exercise assuming that $f$ is a non-degenerated skew-symmetric bilinear form. Is there something I am missing? Thank you for your help!

Martin Sleziak
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jggarita
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  • This doesn't directly answer your question, hence why it is a comment, but this answer to a related question I asked recently might help you (or anyone else with this problem): http://math.stackexchange.com/a/2100056/327486 This question might also be relevant, albeit less so (I would focus on the former): http://math.stackexchange.com/questions/2098559/proof-verification-v-otimes-v-cong-mathscrlv-v/2098828#2098828 Admittedly the proofs/answers given might not help as much if you are not familiar with double-dual spaces. – Chill2Macht Jan 20 '17 at 22:17

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