I'm trying to generalize a method which can solve a problem involving discrete group to continuous group. And the method involve construct a "free vector space" from discrete space. So I'm trying to find a way to construct some kind of "free Hilbert space" from a topological space. But I have no idea how to achieve this goal. Then I find that one can get a free group or free vector space by considering the left adjoint of forgetful functor, so I wonder if one can go this way to get "free Hilbert space". But I don't know how to construct this adjoint.

Let $\mathbf{Hil}$ be the category of Hilbert spaces, $\mathbf{Top}$ be the category of topological spacces, and consider the forgetful functor $ F : \mathbf{Hil} \to \mathbf{Top}.$ Does this functor $F$ have a left adjoint? If it does, what is this left adjoint?

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  • $\begingroup$ I'm sorry I didn't clear about what my question is, I add the problem I had in the question. $\endgroup$
    – Chen
    Oct 3 at 17:55
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1 Answer 1


The left adjoint does not exist. Otherwise, also the forgetful functor $\mathbf{Hilb} \to \mathbf{Top} \to \mathbf{Set}$ to the category of sets had a left adjoint (since left adjoints "compose"), but it doesn't:

Is there a concept of a "free Hilbert space on a set"?

  • $\begingroup$ Thank you for your answer. I think I have to try other way to generalize the method. $\endgroup$
    – Chen
    Oct 3 at 18:11

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