Question

What is a function space (hilbert space, banach space)? Why are infinite dimensions necessary?

Answer 1

a complete (inner product and Norm respectively) vector space.

why not have and infinite number of dimensions ;)

Answer 2

I have read the Wikipedia definitions of functional analysis and function spaces...but I don't understand the intuition and possible uses. I am trying to think of a function space as similar to a vector space but I'm not clear what each axis/dimension looks like (ie. In a 3D vector space, each axis has real numbers.) What does each dimension in a function space have? Functions?

Answer 3

I believe that there are some applications in physics...but I've never studied them.

I took a course in graduate school on functional analysis (though that was a while ago), but that was it. I would suggest you talk to someone who is an expert in functional analysis. I don't think there is anyone here who fits that description.

Answer 4

Thank you.