Hilbert, Banach and Fourier theory

[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

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

 

[mathman]

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

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

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.

 

[center o bass]

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.

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?

 

[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.