Hilbert Space Orthonormal Sets: Alternative to Rudin

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In summary, the conversation is about a course on Hilbert spaces and the use of Rudin's "Real and complex analysis" textbook. The speaker is generally happy with the textbook but has some issues with the section on orthonormal sets. They are looking for suggestions for other textbooks that cover this topic, such as "Introductory Functional Analysis" by Kreyszig, "Foundations of Modern Analysis" by Dieudonne, or books by George Simmons or Sterling K. Berberian. They also mention that Rudin's writing style can be difficult to understand and suggest trying to prove the results oneself before reading a textbook.
  • #1
Hjensen
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I am taking a course on Hilbert spaces and we're using Walter Rudins "Real and complex analysis", which I am generally very happy about.

However, I don't think the section about orthonormal sets (page 82-87) is that nice. In particular, I would like to see a different approach to the theorem 4.18. Does anyone have a suggestion to another text on orthonormal sets/orthonormal bases in a Hilbert space?
 
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  • #2
I would suggest "Introductory Functional Analysis" by Kreyszig. The theorem you want is on page 170.
 
  • #3
a book liked as a student was by edgar lorch, spectral theory. this theorem is on page 68, thm. 3-5, but the proof builds up over several previous pages.

Another good book is Foundations of moden analysis, by Dieudonne, where this material is treated in chapter VI.5.

Anything by George Simmons is also recommended as especially clear.

or introduction to hilbert space by sterling k. berberian. I recall as an undergradutae that I could follow easily every argument in berberian.

Indeed this stuff is available in many places. You might even just try to prove the results yourself, and see how far you get. then read a book to fill in the rest.

In most cases Rudin is my absolute last place to look for something understandably explained. He is one of the few remaining authors (except me sometimes) who seems to take especial pride in being as brief as possiBle instead of being clear.
 
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  • #4
Thanks a lot guys. I'll look into that.
 
  • #5


As a fellow scientist with a background in mathematics, I can understand your desire to explore alternative approaches to understanding Hilbert spaces and orthonormal sets. While Rudin's "Real and Complex Analysis" is a well-respected text, it is always beneficial to explore different perspectives and explanations of mathematical concepts.

One alternative text that I would recommend is "Hilbert Space Methods in Probability and Statistical Inference" by Chrisopher G. Small and Don L. McLeish. This text provides a comprehensive introduction to Hilbert spaces and their applications in probability and statistics. Chapter 2 specifically covers orthonormal bases and their properties in Hilbert spaces, providing a different perspective and potentially a clearer understanding of the topic.

Additionally, "Functional Analysis: An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras" by Joseph Muscat covers orthonormal bases in Hilbert spaces in Chapter 7. This text also includes exercises and examples to reinforce the concepts discussed.

I hope these suggestions will be helpful in your exploration of orthonormal sets in Hilbert spaces. It is always beneficial to seek out alternative resources to deepen our understanding of mathematical concepts. Happy learning!
 

1. What is Hilbert Space Orthonormal Set and how is it different from Rudin's approach?

Hilbert Space Orthonormal Set is a mathematical concept used in functional analysis to describe a set of functions that are orthogonal to each other and have unit norm. This approach is an alternative to Rudin's approach, which uses the concept of inner product spaces to define orthonormal sets.

2. What are the applications of Hilbert Space Orthonormal Sets?

Hilbert Space Orthonormal Sets have various applications in quantum mechanics, signal processing, and image processing. They are also used in the mathematical formulation of quantum mechanics and in the study of Fourier series and transforms.

3. Can you provide an example of a Hilbert Space Orthonormal Set?

One example of a Hilbert Space Orthonormal Set is the set of trigonometric functions (sine and cosine) on the interval [0, 2π]. These functions are orthogonal to each other and have unit norm, making them an orthonormal set in the Hilbert Space of square-integrable functions on the interval.

4. How is the concept of Hilbert Space Orthonormal Sets related to the concept of orthogonality?

Hilbert Space Orthonormal Sets are a special case of orthogonal sets, where the functions not only are orthogonal to each other but also have unit norm. In other words, the inner product of any two functions in a Hilbert Space Orthonormal Set is equal to 0, and the norm of each function is 1.

5. Is there a unique Hilbert Space Orthonormal Set for a given space?

No, there can be multiple Hilbert Space Orthonormal Sets for a given space. However, each set must satisfy the properties of orthogonality and unit norm. The choice of which set to use depends on the specific problem or application at hand.

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