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I am stuck in this exercise.

"Let be $k$ a division ring, and $\textbf{vect-k}$ (resp. $\textbf{k-vect})$ te category of all right (left) vector spaces over $k$ that are of finite dimension, and let D:=$Hom(-, \hspace{0.1cm} _{k}k_{k}):\textbf{vect-k}\rightarrow \textbf{k-vect}$ the usual duality functor (i.e. the controvariant functor that sends every $V_{k}$ in its dual $_{k}V^{*}$ and every linear mapping $f:V_{k}\rightarrow W_{k}$ in its transposed $f^{*}:\hspace{0.1cm }_{k}W^{*}\rightarrow \hspace{0.1cm} _{k}V^{*}$). Show that D is not an isomorphism of categories between $\textbf{vect-k}$ and $(\textbf{k-vect})^{op}$."

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  • $\begingroup$ What are your thoughts on the problem? What have you tried? $\endgroup$ Nov 9, 2019 at 14:39
  • $\begingroup$ I was able to show that D is certainly a duality, i.e. a full, faithful and essentially surjective functor, but I've no idea to get out of this question. An isomorphism of categories means that I can find another functor G such that GF and FG are the identities functors (in their respectively domains), so I think I could suppose that such G exist and then find some contraddiction, but I don't see the contraddiction... $\endgroup$
    – Kushike
    Nov 9, 2019 at 14:47

1 Answer 1

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The obstruction lies in cardinality. Prove that: either $V^*$ is finite-dimensional or of uncountable dimension. Therefore, $(\bullet)^*$ cannot be essentially surjective.

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