# Analysis Are Terry Tao's Books On Analysis Worth Getting?

#### bhobba

Mentor
I learned my analysis from Ruden way back when. It fell apart from overuse so I got two books to replace it - Apostol and Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by Hubbard - both very different in approach. But recently I have read Terry's two volume set is really good using a careful methodological approach. Do people think its worth getting? I don't refer to them as much as when I was teaching myself Rigged Hilbert Spaces and the like, so it would just be to see how a really modern mathematician like Terry would tackle it.

Thanks
Bill

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#### Haborix

Just in the event you are not aware, most of Tao's material is freely, and legally, available (perhaps in less polished form) in some format through his blog. For example, here is the page for Analysis I with a link to the lecture notes it is based on. You can have a look through those and see if you might like it. I can't offer you any expert opinion, my background in analysis is almost identical to your own, but thought I should note the material available through his blog.

#### Demystifier

2018 Award
to see how a really modern mathematician like Terry would tackle it.
I know why is Tao considered a genius, but what exactly makes him modern?

#### stevendaryl

Staff Emeritus
I know why is Tao considered a genius, but what exactly makes him modern?
Because he's still alive (and not too old)?

#### bhobba

Mentor
I know why is Tao considered a genius, but what exactly makes him modern?
He is only 40 - Ruden for example was written in 1953.

Its nothing to do with content - they would contain pretty much the same thing. I have had a look at some of Terry's pages in the book and he seems to concentrate more on the foundations, even addressing one of my favorite issues even my analysis teacher could not answer, S = 1 -1 + 1- 1 ...... = 1 - S ie S = 1/2. Or maybe he could and simply wanted me to investigate it. He doesn't give the full answer - the lectures here do:

The answer is it all depends on your definition of an infinite sum and that any must have certain reasonable properties. There are a number of alternative ones to the usual convergence definition. But the strange thing is, when they give an answer pretty it's much always the same - that they all have the same reasonable properties and that is often, by itself, enough to work out the answer is why.

He just seems to be more 'modern' in approach.

Thanks
Bill

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#### bhobba

Mentor
Because he's still alive (and not too old)?
Basically yes.

But it is a reasonable question. The basics of analysis were worked out by Weierstrass, Cauchy etc in the the 19th century so nothing much has changed content wise - although some books include slightly later topics like Lebesgue measure. So why do you want the view of a current relatively young mathematician? After looking at it he seems to place more emphasis on foundational issues such as Cauchy sequences which he perhaps has found more of value in preparing students for later courses in Hilbert spaces etc.

Thanks
Bill

#### Demystifier

2018 Award
even addressing one of my favorite issues even my analysis teacher could not answer, S = 1 -1 + 1- 1 ...... = 1 - S ie S = 1/2. Or maybe he could and simply wanted me to investigate it. He doesn't give the full answer - the lectures here do:
The answer is it all depends on your definition of an infinite sum and that any must have certain reasonable properties.
He just seems to be more 'modern' in approach.
Hardy wrote a book about such stuff in 1949:
https://www.amazon.com/dp/0821826492/?tag=pfamazon01-20

#### bhobba

Mentor
Well I will be gob smacked. I didn't read it, I figured a bit out for myself and the rest from that lecture series. Maybe my professor wanted me to read that or something similar.

Nice book from the look inside. But Hardy and applied math? Bender is the opposite. Interesting contrast. Then again my professors thought me more pure than applied but gradually I turned and now use things like infinitesimals with gay abandon -- not sure if Terry or its intended audience would approve.

It can get you into trouble. There is a famous theorem that is known as hard to prove - the Feller-Erdös-Pollard theorem. When I was taught it - there was the dreaded - proof omitted. I came up with a elegant simple proof and was proud. Took it along to my professor - he said - good try - but it's wrong - you exchanged the sum of infinite series. Damn. Found the correct proof in Feller later.

Thanks
Bill

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#### stevendaryl

Staff Emeritus
even addressing one of my favorite issues even my analysis teacher could not answer, S = 1 -1 + 1- 1 ...... = 1 - S ie S = 1/2.
How about $1 + 2 + 3 + ... = \frac{-1}{12}$?

#### bhobba

Mentor
How about $1 + 2 + 3 + ... = \frac{-1}{12}$?
Good old Ramanujan Summation:
https://en.wikipedia.org/wiki/1_+_2_+_3_+_4_+_⋯

Cant take credit for that one - already knew it.

Doubt it will last long though - you will easily be able to come up with one I do not know.

It doesnt show 1 - 2 + 3 -4 .......

Here is that one S = 1 -2 +3 - 4 = 1 - (1 - 1 + 1 _ .....) - (1 - 2 +3 - 4.....) = 1 - 1/2 - S.

You can do the rest.

The reason Bender talks about it is exactly as the link says - it has applications in QFT. Ramanujan and QFT - strange combination - or maybe not that strange considering how interconnected math is.

Thanks
Bill

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#### MidgetDwarf

Hmm. Maybe give Bloch: Real Numbers and Real Analysis. It does what Tao's book does but better...

"Are Terry Tao's Books On Analysis Worth Getting?"

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