Functional Analysis (Dover Books on Mathematics) by Frigyes Riesz (PDF)

Description

Classic exposition of modern theories of differentiation and integration and principal problems and methods of handling integral equations and linear functionals and transformations. 1955 edition.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This 1952 book by Frigyes Riesz (1880-1956) and Béla Szőkefalvi-Nagy (1913-1998) is one of my favourite real analysis books because it is so concrete and down-to-earth. It’s a great antidote to some of the very abstract modern treatments of functional analysis.I borrowed this book from the library in 1979 in my first graduate year and was amazed by the Denjoy-Young-Saks theorem (with proof) on pages 17-19, which tells you about the differentiability of completely arbitrary real functions of a real variable. After seeing so many undergraduate theorems which require continuity, or at least measurability, to be able to assert anything interesting, it’s quite a shock to see a differentiability theorem with no preconditions! This theorem follows the statement and proof of Lebesgue’s theorem on pages 5-9, which asserts the differentiability almost everywhere of a general monotonic function. As far as I can tell, there is absolutely no need for any axiom of choice for the proof. However, the proof is really very intricate indeed if you write out all the steps in detail.Then chapter 1 is rounded off with the two fundamental theorems of calculus for the Riemann integral (which is really the Cauchy-Riemann-Darboux integral), and then Darboux’s theorem and a section on functions of bounded variation and the rectification of curves. This is really a lot of fundamental real analysis in just 28 pages.Chapter 2 then covers the Lebesgue integral, L^p spaces and linear functionals in a more or less standard fashion. Chapter 3 presents the Riemann-Stieltjes and Lebesgue-Stieltjes integrals, and the Daniell integral. The remaining chapters cover fairly standard modern topics of functional analysis in a quite concrete down-to-earth way, not in the more abstract way which much later books present functional analysis.PS. 2017-12-4. I should have mentioned that I bought my own copy of Riesz and Sz-Nagy in 2014 via Amazon. So my review is not written entirely from my memories of the borrowed copy in 1979. In fact I have made substantial use of this book in my own writing in the last couple of years.

⭐This book is a classic. All other good books inevitably refer to it in the end for the “too hard” proofs. The text doesn’t burden one with over-generalisations. Most functions are either real or complex valued, for example. The authors of this book are some of the “founding fathers” of the subject, so you are getting it from the horse’s mouth.

⭐Have been interested in functional analysis for some time. This text was a GREAT addition to my collection!!!Riesz gives a very good exposition on the subject – good for both novices to the subject who are at least starting out with a background in linear algebra/analysis and for experts, alike.

⭐The content of this book is good, and, unlike some other translations, the style of the English is appealing.However, the publisher has made the text a little bit small for the paperback.It is bearable, but would be nice to know before purchasing.

⭐good

⭐Uma obra-prima em Análise. Indico como 2a leitura. Isso porquê creio que o conteúdo desse livro só possa ser melhor apreciado após um primeiro contato com o mesmo. O autor trata com maestria a teoria de medida e integração, análise funcional, equações integrais e outras aplicações a teoria de eq dif parciais.私のように、金融工学・ファイナンスに近い領域を仕事にしている数学プロパーでない人のためのレビューです。関数解析の優れた書物は実に多数出版されています。たとえば、Kolmogorov-Fomin『関数解析の基礎』（岩波）、藤田宏他『関数解析』（岩波基礎数学選書）、K. Yosida, Functional Analysis, Splinger、H. Brezis『関数解析』(産業図書)などがあげられます。もちろん、これらは一冊ですべてをカバーするものでなく、一長一短があります。Kolmogorov-Fominと藤田宏他は同程度のレベルで入門書としては最適でしょう。しかし、発展的な議論に関してはやや物足りない。吉田耕作先生の本は名著ですが、半群を勉強するのでなければ迂遠な感を否めない。BrezisはSobolev空間などの発展的な議論に関しては大いに参考になりますが、線形空間からの導入に関してやや難があります。さて、本書です。金融工学では「Rieszの表現定理」が重要な役割を果たす部分があります。それ自体は直交射影定理の拡張ですから別に難解なものではありませんが、Riesz自体はこれをどのように捉えていたかに長らく興味がありました。本書では、L2空間のユニタリ塊換であるBochnerの定理との関連で展開されていることに興味が惹かれました。ベクトル束の概念を提示したRieszのことですから、線形空間から関数空間を構成する導入にも怠りがありません。英語であろうと日本語であろうと、数学を勉強する際にはあまり関係がありませんから、Rieszの表現定理を勉強しようという数学プロパーでない人も本書を一読されることをお勧めします。Es kann nicht die Aufgabe sein, das Buch zu rezensieren. Die Einzige Leistung, die Amazon und der Verkäufer erbringen ist die Lieferung des Buches zu dem vereinbarten Preis. Eine Fachrezension gehört in die Fachzeitschrift. Ich mache hier keine Reklame für das Buch!! obwohl es mir sehr gut gefällt. Das ist allein meine Sache. Damit muss sich Amazon endlich abfinden. Den Einriss im Deckel kann ich mit Tesafilm stoppen. Das schadet dem lesbaren Inhalt.

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