What's the connection between Banach/Hilbert spaces and tools like power series, Fourier transforms etc. 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?
 A: 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.
A: The space $\ell_2$ is the space of infinity sequences square additive.
You can map a vector in $\ell_2$, for example $(\xi_1, \xi_2  ,\cdots, \xi_k \, \cdots)$ to a power series $\xi_1 + \xi_2 z + \cdots, \xi_k z^k + ...$.
This is known as the Z transform. think about $z=\exp(i \omega \Delta  t)$ where $\omega = 2 \pi f$, and $f=1/\Delta t$ is the frequency. It is in this way that a power series is a Fourier series. If you have a finite
sequence, then there is a maximum frequency (Nyquist frequency $f=1/(2 \Delta t)$ and the series is a DFT or Discrete Fourier transform. The
signal in time is periodic.
This links  Hilbert spaces $\ell_2$, with power series and Fourier transform. 
In the continuum (that is, for example $L_2$) there is no power series. As
$\Delta \to 0$ the sum become an integral and we are talking about Fourier transforms pairs $f(t) = \frac{1}{2 \pi} \int_0^{\infty} F(\omega) \exp(-i \omega t) d \omega $, $F(\omega) = \int_0^{\infty} f(t) \exp(i \omega t) dt$.
Here is how you can think about these integrals. If a set $\{ \phi_i \}$ is an ortho-normal basis in $L_2$ then , any function $f(t) \in L_2$
can be written as:
$f(t) = \sum_i c_i \phi_i $
The coefficients $c_i$ are known as Fourier coefficients. Since
$\phi_i$'s are orthonormal,
$c_i = \langle f \, , \, \phi_i \rangle$. These are known as
Fourier coefficients. The orthonormal basis are
$\phi(t) = \exp(i \omega t)$ (up to a normalization factor). Think of $c_i$ as the frequency component $F(\omega)$, and the inner product is the integral 
for $F(\omega)$ above. The derivation of the $f(t)$ integral
comes by plugging the coefficient $c_i$ into the sum for
$f(t)$ and finding that $f(t)$ is in both sides, then there got
to be a Dirac delta that will take you to the integral representation
for $f(t)$. If you want details I can elaborte.-
