I was trained as an engineer, where the beauty of the foundational math is often brushed away - or interpreted in a 'useful' way. I am now learning about quantum computation, which means learning more of how physicists see it.
Suppose we look at the space of well-behaved, continuous curves on the interval $[0,1]$. As an engineer, you are told this is an infinte-dimensional space, precisely because any such function has a Fourier transform, and the infinite number of different Fourier terms form the basis.
A physics/quantum text I looked at argued thus: think of the value at any point $x$ that is in $[0,1]$. In fact, just as we usually give subscripts for bases ($i, j,k$), imagine $x$ being the subscript - and then each point $x$ is a 'dimension' and the space is infinite-dimensional.
Maybe there is something wrong with that, but I note that it meets the basics we are looking for, including the fact that the inner product, $<f,g>=\int f(x)\cdot g(x) dx$ sure looks like the limiting case of the usual dot product, but extended to infinite dimensions. I also note that if you want every function to be expressible as the infinite sum of projections/inner products, it seems to work so long as you allow $\delta (x)$ to be used as the bases.
What is the more rigorous approach to what this infinite-dimensional (Hilbert) space really is based upon? In what way is it infinite-dimensional? Are both or either of these views right, and why or why not?
I note, by the way, that if you take the view the book I am reading takes (what I will call the 'axis approach') then that space always has a dimension that is uncountably infinite, because the points on the real line are all indexes. The same would be true for the Fourier transform view...except for this asymmetry: for functions that are periodic on $[0,1]$, the 'axis' view is still uncontably infinite in dimension, but the Fourier view is now countably infinite.