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.

 

[blue_raver22]

Hello,

Hilbert, Banach, and Fourier theory are all important areas in mathematics that have many applications in various fields, including physics, engineering, and signal processing.

To get a quick overview of Hilbert spaces, I would recommend starting with a general textbook on functional analysis, such as "Functional Analysis" by Walter Rudin or "Introductory Functional Analysis with Applications" by Erwin Kreyszig. These books cover the necessary results of Hilbert and Banach spaces and also include applications to Fourier theory. Additionally, there are many online resources and lecture notes available on the topic that can provide a quick overview.

The virtue of a Hilbert space being defined as a Banach space is that it allows for the use of the powerful tools and techniques from functional analysis, such as the Hahn-Banach theorem and the Banach-Steinhaus theorem, to study the properties of Hilbert spaces. These properties are crucial in understanding the convergence and completeness of Hilbert spaces, which are essential for the study of Fourier series and transforms.

To prove that $L^2 (S^1)$ is a Banach space with the standard Hilbert space norm, one needs to show that it satisfies the three properties of a Banach space: completeness, linearity, and norm. Completeness can be shown by using the fact that $L^2 (S^1)$ is a closed subspace of the space of all square-integrable functions on $S^1$. Linearity follows from the linearity of the integral, and the norm property can be proved using the Cauchy-Schwarz inequality. Overall, the completeness and norm properties of $L^2 (S^1)$ make it a suitable space for studying Fourier series and transforms.

I hope this helps to provide a better understanding of the theory of Hilbert spaces and its connection to Fourier theory. If you have any further questions, please feel free to ask.