# Left adjoint to the forgetful functor from Hilbert spaces to topological spaces

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?

• Quick beginner guide for asking a well-received question + please avoid "no clue" questions: include some work in your post. Oct 3 at 17:40
• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
– Community Bot
Oct 3 at 17:51
• I'm sorry I didn't clear about what my question is, I add the problem I had in the question.
– Chen
Oct 3 at 17:55
• @Chen Ignore the bot. Oct 3 at 17:57
• Your question was already clear enough. What was/is lacking is some thoughts and/or attempts. Oct 3 at 17:59

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: