Name: A radical approach to real analysis
ISBN: 9780883857472

A radical approach to real analysis

"In the second edition of this MAA classic, exploration continues to be an essential component. More than 60 new exercises have been added, and the chapters on infinite summations, differentiability and continuity, and convergence of infinite series have been reorganized to make it easier to identify the key ideas. A Radical Approach to Real Analysis is an introduction to real analysis, rooted in and informed by the historical issues that shaped its development. It can be used as a textbook, or as a resource for the instructor who prefers to teach a traditional course, or as a resource for the student who has been through a traditional course yet still does not understand what real analysis is about and why it was created"--Publisher description.

What Hacker News says

fofoniMar 2018

Hi, I'm a maths student in Brazil myself :) I'll answer in english, though, since I'm not sure about the HN policy on comments in foreign languages.

Can I ask you which university you're from? I'm not aware of many universities besides UFRJ which offer an "Applied Maths" degree.

About real analysis, I took the summer course in IMPA, an I used only Elon's books. I really like them (for the books themselves, not just because they're in portuguese). There are two: the thick one: "Curso de Análise", and the thin, silver one: "Análise Real" (I like to call them Elão and Elinho :)). Elão is very detailed and has lots of examples, but mentions topics which may be too specific and not covered in your course. Elinho is much more terse, and great if you need a quick summary.

I would also consider reading David Bressod's "A Radical Approach to Real Analysis". It's an awesome book, which mentions historical motivations for everything, and has a really different approach to teaching analysis (it will certainly help you learn analysis, but might not help too much in your course, since it's quite non-traditional).

If you're not used to proof techniques, I highly recommend Keith Devlin's "Introduction to Mathematical Thinking".

About strategies to studying analysis: examples. I think it's really important to work out lots of examples by hand all the time. Every time you read a definition in your textbook, whatever it is, close the book and try to think of some examples of mathematical objects which satisfy the definition. When you're done, try to think of other examples which differ significantly from the ones you came up with before. When you open your book again, if the author presents examples, read them with attention. TLDR: as the other comments have made it very clear, you shouldn't be reading an analysis book without a pencil on your hand; you should feel active, not passive, while studying analysis.

Finally, I don't know about any youtube channels that could help you with an analysis course, but you should be aware of Mathematics.StackExchange. It's a great Q&A website/community; I've asked a lot of questions there while studying for my undergrad courses.

Wish you the best in your course and you maths career :)

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hyperbovineOct 2010

One of my favorite not-much-talked about analysis books is "A Radical Approach to Real Analysis" by Bressoud. Must read if you enjoy reading about the history of mathematics and famous mathematicians. A lot of people here recommend Rudin or Royden, but I have seen many people become turned off to the whole field of analysis because of how terse and user-unfriendly those are. In that sense, ARTRA is the polar opposite. The follow-up, A Radical Approach to Lebesgue's Theory of Integration, is also superb.

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bkovitzJun 2008

Yup, motivation is the blocker.

I have repeatedly gotten tired of math when it seemed like meaningless puzzle-solving. Why do I really care, for example, how many nonabelian groups of order 36 exist? When math seems to me like it's providing a vocabulary for framing and answering deep questions about the world, or making it computationally feasible to find the stamp of causal influence in data or design real things that are impossible to make without out, then the motivation is there.

This might just be an artifact of the way math is often taught. For example, real analysis is often taught as meaningless proof-finding (e.g. proving things that seem either pointless or obvious). But there's a fantastic book, _A Radical Approach to Real Analysis_ by David Bressoud, that teaches the exact same subject matter as the fruit of deep, pressing, and non-obvious questions that stirred debate among mathematicians for around 100 years.

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nextosSep 2022

For example, calculus uses notation and terminology that predates the modern limit-based field Weierstraß and others built. It's really confusing [1].

Statistics is even worse. A mix of old tricks developed to avoid computations when these were expensive. See [2].

[1] A Radical Approach to Real Analysis https://www.davidbressoud.org/aratra/

[2] The Introductory Statistics Course: A Ptolemaic Curriculum? https://escholarship.org/uc/item/6hb3k0nz

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WoahNounJul 2021

This article reminds me of the story of Fourier. It's hard to tell the history of math without the way in which Fourier series and the heat equation shaped the development and increasing rigor in calculus. The mathematicians of the day were extremely skeptical of infinite sums of trigonometric functions because it didn't fit their "a-priori" model of calculus.

>Here was the heart of the crisis. Infinite sums of trigonometric functions had appeared before. Daniel Bernoulli (1700-1782) proposed such sums in 1753 as solutions to the problem of modeling the vibrating string. They had been dismissed by the. greatest mathematician of the time, Leonhard Euler (1707-1783). Perhaps Euler scented the danger they presented to his understanding of calculus. The committee that reviewed Fourier's manuscript: Pierre Simon Laplace (1749-1827), Joseph Louis Lagrange (1736-1813), Sylvestre Francois Lacroix (1765-1843), and Gaspard Monge (1746-1818), echoed Euler's dismissal in an unenthusiastic summary written by Simeon Denis Poisson (1781-1840). Lagrange was later to make his objections explicit.

>Well into the 1820s, Fourier series would remain suspect because they contradicted the established wisdom about the nature of functions. Fourier did more than suggest that the solution to the heat equation lay in his trigonometric series. He gave a simple and practical means of finding those coefficients, the ai, for any function. In so doing, he produced a vast array of verifiable solutions to specific problems. Bernoulli's proposition could be debated endlessly with little effect for it was only theoretical. Fourier was modeling actual physical phenomena. His solution could not be rejected without forcing the question of why it seemed to work.[0]

I picture this in my head as Fourier setting some shit on fire, hand calculating Fourier coefficients, then just pointing and yelling "SEE! SEE!" at Poisson and Lagrange.

[0]: A Radical Approach to Real Analysis - David Bressoud

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