We know that if you have a map $f: A \to H $ from a set $A$ to the unit sphere of a Hilbert space $H$, you can get a bounded linear map $\tilde{f}: L^2(A) \to H$. This construction can be view as considering the left adjoint to the forgetful functor from the category of Hilbert spaces to the category of weighted sets which is showed in the anwser https://math.stackexchange.com/q/3625350 .
I wander if one can do the same thing in the continuous situation. To be specific, if there is a continuous map $ g: X \to H $ from a topological space $X$ to the unit sphere of a Hilbert space $H$, can you still get a bounded linear map $\tilde{g}: H_{X} \to H$, where $H_{X}$ is some Hilbert space depending on topological space $X$ ?
First I try to replace the sum of elements in $L^2(A)$ in discrete situation to some kind of integral. To be specific, I try to define $H_{X}$ as $L^2(X)$ and $\tilde{g}$ as $ \tilde{g}(\varphi(x)):=\int_{X} \varphi(x)g(x)dx $. But in the problem I'm dealing with, it may be difficult to find a nature measure on the topological space $X$ . Then I want to know if I can get some idea from the view of category theory. But it seems not trivial to consider the idea of "free Hilbert spaces", even in the discrete situation you have to consider "the category of weighted sets" to avoid the unbounded situation.
So my questions are:
- Is there some kind of " the category of weighted topological spaces" to allow you define "bounded map" from a topological space to a Hilbert space in the view of category theory.
- If the answer of the first question is yes, does the forgetful functor from the category of Hilbert spaces to "the category of weighted topological space" have a left adjoint? If it has, what is the left adjoint?