I'm trying to define "Analytic function".
I want a definition that covers all interesting cases.
To be specific, let me explain what exactly I want
Here is the definition of analytic function in wikipedia:
Let $D$ be an open subset of $\mathbb{R}$.
Let $f:D\rightarrow \mathbb{R}$ be a function.
If for every $x_0\in D, \exists$ a neighborhood $N$ of $x_0$ such that $\forall x\in N\cap D, f(x)=\sum c_n (x-x_0)^n$ ($c_n$ is real), then $f$ is said to be analytic on $D$.
I know that every holomorphic function is analytic(in complex sense). However, I want to make the term "analytic" covers wider cases.
So here is a definition I figured out:
Let $(V,\| \cdot \|)$ be a Banach space over $\mathbb{K}$.
Let $D$ be an open subset of $\mathbb{K}$.
Let $f:D\rightarrow V$ be a function.
If for every $x_0\in D, \exists$ a neighborhood $N$ of $x_0$ such that $\forall x\in N\cap D, f(x)=\sum c_n (x-x_0)^n$ ($c_n\in V$), then $f$ is said to be analytic on $D$.
However, I don't know whether this definition makes the term "analytic" fruitful. That is, if a function $f:D\rightarrow V$ is analytic in the above sense, then is $f$ infinitely differentiable?
If not, what would be a definition of analytic function that covers many interesting cases?
Yes, the definition works and is fruitful (it helps with the study of the spectrum of an operator, since $\lambda\mapsto (\lambda I-T)^{-1}$ is analytic where defined). Note that the real case can be reduced to the complex case by introducing the complexification of $V$. Banach-space-valued analytic functions are well studied (Google search would bring up a bunch of sources). The notes Holomorphic vector-valued functions by Paul Garrett contain a proof that a convergent power series with coefficients in a Banach space is infinitely differentiable: see the very last corollary.
In fact, the fundamentals of complex analysis carry over to Banach-space-valued setting so well that one is led to ask: Where does the theory of Banach space-valued holomorphic functions differ from the classical treatment?