An over-zigzag (if not silly) writing of definition of analyticity in Amann's Analysis Book.

Question

In a famous textbook, Amann's Analysis I, the author introduce the analyticity as the below picture. Notice that the author defined this terminology with respect to a set $D$, rather than a point $c$. Actually, there is no a definition for being able to say something like "a function $f$ is analytic at a point $c$" in this book.

And then, in Remarks (c), the author said that "analyticity" is a local property; that is, for $f:E\to\Bbb R$, if ($\forall x\in E,~f$ is analytic on a neighborhood of $x$), then $f$ is analytic on the whole domain $E$. Let's stop here, and look at how the same thing is discussed in Terrence Tao's Analysis I:

The difference is that Tao first defined what is called analytic at a point $c$, then simply extend to what is called analytic on a set $E$, which is more straightforward and intuitive. Actually, Amann didn't really get rid of defining the "one point" version. Look closer at Amann's definition, his definition is essentially equivalent to say $f$ is called analytic on $D$ if for each $x_o\in D$, $f$ is "somewhat analytic at $x_0$", the remaing all words is his original definition that I omitted is just the definition of "$f$ is somewhat analytic at $x_0$". So I think the way he write is quite zigzag and unnatural, if not silly. Why not give the version of one-point analytic first? On the other hand, if he had stated the "one point" definition first, then "set version", then his Remark (c) might been at least quite easy enough, if not too trivial. Am I correct? Or is there other reason that he chose so?

Answer 1

Actually I find Amann's version more natural, because unlike continuity, differentiability, and even smoothness, the set of points where $f$ is analytic at is always an open set.

(also there is a sort of trap where if you have a function that is $D^{k+1}$ at a point then it implies that it must be $C^{k}$ on some open set around that point, but when talking about smoothness there are functions that are smooth at a point but not smooth on any open around it, which is a bit unintuitive if you compare with the previous scenario)

One point that is not made very clear in either definition is that if you have a power series $\sum a_nx^n$ with a positive radius of convergence $R > 0$, then if $|y|<R$, then writing $\sum a_nx^n = \sum a_n((x-y)+y)^n$, then you can use the Binomial theorem and reorder terms to get a new power series $\sum b_n (x-y)^n$ that converges for $|x-y| < R-|y|$. And so, if a function is analytic somewhere, truely it is analytic on an open set around it.

With Amann's definition, this fact is pushed on the side of one that has to show analyticity (i.e. it's hard to see but one has to prove the existence of the power series on a whole open set) so it is harder to show that a given function is analytic, however if you are on the side of someone using analyticity, the existence of power series is given for free.

Tao's definition is the reverse, it makes proving analyticity easier, but in exchange it is "less useful" as in it is not obvious that power series exist everywhere.

Amann's definition kinda internalizes this theorem about power series, and assuming you have lots of theorems to prove analyticity (for examples that polynomials are analytic, that compositions of analytic functions are analytic, that $\exp$ is analytic etc) that are important no matter the version of the definition at your disposal, you may not even notice that it's harder to prove, but you will still get to enjoy the added usefulness without having to mention an extra theorem everytime.