Relationship between Noether's First Isomorphism Theorem and Dual Spaces

Question

I have been studying about these topics. Noether's First isomorphism theorem states that if $f:V\to W$ is linear, then there exists an isomorphism between $V/\text{Ker}(f)$ and the subspace $\text{Im}(f)$. I have seen in some books how they show that $V$ and $V^*$ are isomorphic in finite dimension using the bases and the fact that they have the same dimension. But is there an alternative way to show the isomorphism of $V$ and $V^*$ using Noether's Theorem? Both results make me believe that they have some relation, but I am not sure, and I can't find any information about it either.

Answer 1

No, there's no way to show $V \cong V^*$ using the first isomorphism theorem -- at least, not without involving dimension somehow. The reason for this is quite simple: the first isomorphism theorem holds for any and all vector spaces, whether finite-dimensional or infinite-dimensional (in fact, it holds much more widely: there is a version of this theorem for groups, monoids, rings, ... - it can be shown that it holds in all equational theories).

But $V \cong V^*$ only ever holds if $V$ is finite-dimensional! See https://en.wikipedia.org/wiki/Dual_space#Infinite-dimensional_case for examples, and Why are vector spaces not isomorphic to their duals? for further discussion.