# Hilbert, Banach and Fourier theory

• center o bass
In summary, the conversation discusses the need for understanding Hilbert spaces in relation to Fourier series and transforms at a higher level. The individual has a few questions regarding recommended literature, the virtue of Hilbert spaces as Banach spaces, and the proof of completeness in the norm for a specific example of a Hilbert space. The summary also mentions that the most important theorems in Banach spaces are those that guarantee square integrability, which is a crucial property in quantum mechanics.
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

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

center o bass said:
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.

mathman said:
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?

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.

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.

## 1. What is Hilbert theory?

Hilbert theory is a branch of mathematics that deals with the study of Hilbert spaces, which are mathematical structures used to model infinite-dimensional vector spaces. It was developed by the German mathematician David Hilbert in the late 19th and early 20th centuries.

## 2. What is Banach theory?

Banach theory is a branch of functional analysis that focuses on the study of Banach spaces, which are complete normed vector spaces. It was developed by the Polish mathematician Stefan Banach in the early 20th century.

## 3. What is Fourier theory?

Fourier theory, also known as Fourier analysis, is a branch of mathematics that deals with the study of periodic functions and the representation of functions as sums of simpler trigonometric functions. It was developed by the French mathematician Joseph Fourier in the early 19th century.

## 4. How are Hilbert, Banach, and Fourier theories related?

Hilbert, Banach, and Fourier theories are all branches of mathematics that deal with the study of infinite-dimensional spaces and functions. They are related in that they all use similar concepts and techniques, such as the notion of completeness and the use of orthogonal functions.

## 5. What are the applications of Hilbert, Banach, and Fourier theories?

These theories have numerous applications in various fields, such as physics, engineering, and signal processing. For example, Hilbert spaces are used in quantum mechanics to describe the state of a physical system, Banach spaces are used in functional analysis to study differential equations, and Fourier analysis is used in image and signal processing to decompose signals into different frequency components.

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