Let $k$ be a field. I don’t know the conventional notation of the category of inner product spaces over field $k$, donate it $\mathbf{IPrd}_k$ here. The question is, does the functor $$U: \mathbf{IPrd}_k\to \mathbf{Vect}_k$$ which forgets the inner product structure have a left adjoint?
I suspect if it is possible to define a inner product on each vector space naturally, for if so, one could further define a natural isomorphism from each vector space to its dual. So I guess that either the adjunction doesn’t exist, or the inner product is defined in some vector spaces quite different than the original one.
EDIT: By the time I asked this question, I didn’t realized that there are underlying troubles as commented below. For clarity, let’s assume that what it meant was that morphisms are linear maps, and $k$ is a subfield of $\mathbb C$.
Of course this is much weaker than the definition of $\mathbf{Hilb}$. I wish to know either
- the answer to the question, or
- why this idea doesn’t work.
My apologies for this kinda problematic question.