I've learned, abstractly, about Banach and Hilbert spaces, and more concretely about $l^p$ and $L^p$ spaces. I also understand these ideas have something to do with a variety of tools that are useful in physics and engineering, like Power series, Fourier transforms, etc. But no one has really ever spelled out the connection to me.
This is probably a dumb question, but what's the connection?
For example, $L^2$ is a very natural domain for the fourier transform, because it's an isomorphism on that space. In other words, the fourier transform is linear, the fourier transform of every function in $L^2$ is also in $L^2$, every function in $L^2$ is the fourier-transform of some other function in $L^2$, and the $L^2$-norm of a function and its fourier transform is always the same. Oh, and it's one-to-one also.