Real analysis is very different from calculus.

How to learn real analysis?

Thanks

water wrote:
> Real analysis is very different from calculus.
>
> How to learn real analysis?
>
> Thanks


Let me describe an exercise I am doing now, for me to rehearse rather than to learn basic
real analysis. But it may well be a kind of introduction.

An important pastime of mine is making visualisations to mathematical subjects.
One of my favourites is the exploration of some Fourier series and their partial sums.

Consider this Fourier series of a block waveform:

Series A1: sin(t)/1 + sin(3t)/3 + sin(5t)/5 + ...

This series converges to pi/4 for 0 < t < pi (mod 2.pi), to -pi/4 for pi < t < 2.pi (mod
2.pi), and to zero for t = 0 (mod 2.pi) and t = pi (mod 2.pi).

Now take a look at the graphics of the successive partial sums. You will of course observe
the period 2.pi and you will see that all of them are odd functions of t.
So we can concentrate on [0, pi].

As you take more and more terms the graphic more and more resembles the block wave.
However, the waviness continues to exist. Looking closely you guess that the wavelets get
shorter and lower in a fixed proportion. Observe what happens closely to pi/2, say from 80
to 100 degrees. You guess that the maximal slope of the tangent to the waveform never gets
below 1.

It requires no more than elementary algebra, goniometry and calculus to confirm what you
observed in the graphics of the partial sums.
BTW, it is very instructive to study the derivative series

Series A0: cos(t) + cos(3t) + cos(5t)

and the antiderivative series

Series A2: - (cos(t) + cos(3t)/9 + cos(5t)/25 + ... )

along with the original series A1.


Now the questions...

(A) The series 1/1 + 1/3 + 1/5 + ... diverges. How come that the original sine series
converges for all arguments? There seems to exist a strange discontinuity at t=0 ...?!

(B) What about the waviness of series A2?

(C) What is the exact relation of the more or less wild behaviours of the partial sums to
the rate of decrease of the successive coefficients?

Etc, etc.

In my opinion these three Fourier series offer an excellent point of departure for
studying real analysis. To mention a few topics that are immediately needed in this
specific situation: limits and differentiability; metric spaces; Hilbert spaces;
Dirichlet's theorem on Fourier series.

Some books I can recommend:
Dieudonné: Modern analysis;
Walter Rudin: Real and complex analysis;
Walter Rudin: Functional analysis;
Riesz and Nagy: Leçons d'analyse fonctionnelle

Happy studies: Johan E. Mebius

On Mon, 11 Feb 2008 11:39:37 +0100, JEMebius
<jemebius*xs4all.nl> wrote:

>Some books I can recommend:
>Dieudonné: Modern analysis;
>Walter Rudin: Real and complex analysis;
>Walter Rudin: Functional analysis;
>Riesz and Nagy: Leçons d'analyse fonctionnelle

N.B. None of these is for beginners!

At an introductory level, Spivak's "Calculus" gets a lot of
recommendations, and the third edition was reprinted by
Cambridge University Press in 2006 (at a reasonable price).
(I really must read it myself ...)

An almost arbitrary list of a few others worth considering;

Apostol, "Mathematical Analysis" (out of print)
Beardon, "Limits: A New Approach to Real Analysis"
Bressoud, "A Radical Approach to Real Analysis"
Burkill, "A First Course in Mathematical Analysis"
Carothers, "Real Analysis"
DePree & Swartz, "Introduction to Real Analysis" (expensive!)
Rudin, "Principles of Mathematical Analysis"
Simmons, "Introduction to Topology and Modern Analysis"
(maybe a second course? - it's been a while since I saw it)

Supplementary reading:

Gelbaum & Olmsted, "Counterexamples in Analysis"

--
Angus Rodgers
(twirlip* eats spam; reply to angusrod*)
Contains mild peril

On Mon, 11 Feb 2008 14:33:28 +0000, Angus Rodgers
<twirlip*bigfoot.com> wrote:

>On Mon, 11 Feb 2008 11:39:37 +0100, JEMebius
><jemebius*xs4all.nl> wrote:
[snip]

>At an introductory level, Spivak's "Calculus" gets a lot of
>recommendations, and the third edition was reprinted by
>Cambridge University Press in 2006 (at a reasonable price).
>(I really must read it myself ...)

IMHO, Courant, or the later revision by Courant and John, offers more
than Spivak. But, since you mentioned the third edition of Spivak,
you might want to consider the upcoming fourth. See
<http://www.mathpop.com/mainhtms/bip.htm>


For those that haven't already seen it, a wonderful book with a fresh
perspective is "Introduction to Calculus and Classical Analysis" by
Omar Hijab, ISBN-13 978-0387693156. I rather enjoyed this -
<http://www.math.rutgers.edu/~zeilberg/Opinion27.html>

which links to
<http://www.math.rutgers.edu/~zeilberg/MarvinKnopp>

Instead of saying,
>
> Series A2: - (cos(t) + cos(3t)/9 + cos(5t)/25 + ... )
>


Would it also be possible to say

"The sum of cos(((x-1)2)+1)/(((x-1)2)+1) from 1 to infinity" ?

Master Fruity Loops wrote:
> Instead of saying,
>> Series A2: - (cos(t) + cos(3t)/9 + cos(5t)/25 + ... )
>>
>
>
> Would it also be possible to say
>
> "The sum of cos(((x-1)2)+1)/(((x-1)2)+1) from 1 to infinity" ?


Yes, of course. But please say it then correctly.
What did you do to the argument "t"? You simply forgot it? And where are the correct
denominators? Or did you simply reply too fast? Exercise in fast typewriting gone wrong?

I guess you intend to let the variable "x" run through the positive integers so as to
obtain the coefficients 1, 3, 5, ..., which appear in the original series.
At this place I would prefer "N" or some other character from the range i...n to "x".

A formula without "..." reads

minus Sum from N = 1 to infinity of cos([2N-1]t)/[2N-1]^2

But in newsgroup posts I prefer easy readability to utmost formal correctness.

Ciao: Johan E. Mebius

"JEMebius" <jemebius*xs4all.nl> wrote in message
news:47B025E9.7050409*xs4all.nl...
> water wrote:
>> Real analysis is very different from calculus.
>>
>> How to learn real analysis?
>>
>> Thanks
snip
>
> Some books I can recommend:
> Dieudonné: Modern analysis;
> Walter Rudin: Real and complex analysis;
> Walter Rudin: Functional analysis;
> Riesz and Nagy: Leçons d'analyse fonctionnelle
>
> Happy studies: Johan E. Mebius

This one by Bruckner, Bruckner, & Thompson is free and appears to be
well-written:
http://classicalrealanalysis.com/download.aspx

(there are 3 books there, the first link on the page, Elementary Real
Analysis, is the one you want)

> >> How to learn real analysis?
...

> This one by Bruckner, Bruckner, & Thompson is free and appears to be
> well-written:http://classicalrealanalysis.com/download.aspx
>
> (there are 3 books there, the first link on the page, Elementary Real
> Analysis, is the one you want) ...


Actually the discussion here is taking the reader through a pretty
broad range of real
analysis levels. If you are following this thread as a relative
novice then make sure to
start at the beginning. We have three real analysis texts on our web
site. The Elementary
Real Analysis [TBB] starts at the calculus level and goes pretty far
(short of measure theory
though). The TBB-Dripped (for the adventurous) does the same but with
the modern integration
theory added in. Then [BBT] covers material that is commonly in a
graduate real analysis course
[at least graduate level in north america--well maybe freshman in
russia and hungary :-) ]

All three books are free hyperlinked PDF files, designed for on
screen viewing. That's my design
and could certainly be improved no doubt if anyone has some advice.
Trade paperback versions
will soon be available for TBB and, later, for BBT. Our feedback
from a class of 20 in the midwest
who were using [TBB] was that, while they appreciated the free PDF
files, they had all found used
copies on the internet that they wanted to use as well. For that
reason we are working on the paperback
versions.

All information on the texts is at http://www.ClassicalRealAnalysis.com.
Or my blog http://classicalrealanalysis.blogspot.com.

<briansthomson*gmail.com> wrote in message
news:9d8a077f-5725-4fec-9f0b-e49d11b660a7*s8g2000prg.googlegroups.com...
> > >> How to learn real analysis?
> ...
>
>> This one by Bruckner, Bruckner, & Thompson is free and appears to be
>> well-written:http://classicalrealanalysis.com/download.aspx
>>
>> (there are 3 books there, the first link on the page, Elementary Real
>> Analysis, is the one you want) ...
>
>
> Actually the discussion here is taking the reader through a pretty
> broad range of real
> analysis levels. If you are following this thread as a relative
> novice then make sure to
> start at the beginning. We have three real analysis texts on our web
> site. The Elementary
> Real Analysis [TBB] starts at the calculus level and goes pretty far
> (short of measure theory
> though). The TBB-Dripped (for the adventurous) does the same but with
> the modern integration
> theory added in. Then [BBT] covers material that is commonly in a
> graduate real analysis course
> [at least graduate level in north america--well maybe freshman in
> russia and hungary :-) ]
>
> All three books are free hyperlinked PDF files, designed for on
> screen viewing. That's my design
> and could certainly be improved no doubt if anyone has some advice.
> Trade paperback versions
> will soon be available for TBB and, later, for BBT. Our feedback
> from a class of 20 in the midwest
> who were using [TBB] was that, while they appreciated the free PDF
> files, they had all found used
> copies on the internet that they wanted to use as well. For that
> reason we are working on the paperback
> versions.
>
> All information on the texts is at http://www.ClassicalRealAnalysis.com.
> Or my blog http://classicalrealanalysis.blogspot.com.


FWIW, I read through the first chapter of Elementary Real Analysis and
thought that it was very well presented. I also have Georgi Shilov's book
on Real Analysis and I like his writing, but thought that your text was
easier to comprehend. I just finished Calc III in December and am early
into a class on Differential Equations, so Real Analysis is my first "real"
mathematics rather than just computational stuff.

Therefore, it was nice to find that your presentation was clear and to the
point, not assuming a lot of underlying knowledge or jumping around. In
contrast, my Differential Equations text by Boyce & Brannan seems to
constantly be assuming other knowledge as they go from one formula to the
next with little or no explanation and simply assume that the steps are
obvious. It is a tedious tome, but BBT is written the way a math book
should be written IMHO.
Alan