Hilbert Space Orthonormal Sets: Alternative to Rudin

[Hjensen]

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?

 

[micromass]

I would suggest "Introductory Functional Analysis" by Kreyszig. The theorem you want is on page 170.

 

[mathwonk]

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.

 

[Hjensen]

Thanks a lot guys. I'll look into that.

 

[CellCoree]

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!