# Hilbert, Banach and Fourier theory

1. Feb 20, 2014

### center o bass

Hi. I want to get a quick overview of the theory of Hilbert spaces in order to understand fourier series and transforms at a higher level. I have a couple of questions I hoped someone could help me with.

- First of all: Can anyone recommend any literature, notes etc.. which go through the sufficient results of Hilbert and Banach spaces (in a quick way) with the aim to understand Fourier theory?

-What is the virtue of a Hilbert space being defined as a Banach space, i.e. a normed linear space in which any Cauchy sequence of elements converge to an element in the space?

- How do I prove that $L^2 (S^1)$ where $S^1$ is the circle is a Banach space with the standard hilbert space norm

$$\langle f, g\rangle = \int_{S^1} f^*(x) g(x) dx?$$

2. Feb 20, 2014

### mathman

Last two questions:

Since a Hilbert space is a Banach space, theorems about Banach spaces automatically apply to Hilbert space.

For the last question, you need to prove completeness in the norm.

3. Feb 21, 2014

### center o bass

What are the most important theorems we care about in Banach spaces?

For example in QM square integrability is a very important property. What theorems guarantee square integrability?

4. Feb 21, 2014

### mathman

Banach spaces theorems are about Banach spaces. Your question (guarantee square integrability) is something that has to be shown to justify calling a set a Hilbert space. Once that has been shown, then you can use Hilbert space theory and Banach space theory to ascertain certain properties.