---
tag: B538T38WDS
kind: definition
name: Analytic function between Banach spaces
msc: [46G20, "26E05"]
defines: H1KHRWNH8B
uses: [06GZ956657, 0443XNARDV, 06EKW1AKXF, 081WKEDK6C, 09RTYYHXKA]
sources:
  - url: https://en.wikipedia.org/wiki/Analytic_function
---

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $X$ and $Y$ be [[Banach spaces|06GZ956657]] over $\mathbb{K}$.
For $n \ge 1$, let $L^n(X, Y)$ be the space of continuous [[multilinear maps|0443XNARDV]] $A \colon X^n \to Y$, with the norm $\norm{A} = \sup \set{\norm{A(h_1, \ldots, h_n)} : \norm{h_1}, \ldots, \norm{h_n} \le 1}$.
Let $L^0(X, Y) = Y$.
Let $U \subseteq X$ be open, let $f \colon U \to Y$ and let $a \in U$.
The map $f$ is *analytic at $a$* if there are $A_n \in L^n(X, Y)$ for $n \ge 0$ and a real $r > 0$ such that:

1. $\sup_n \norm{A_n} r^n < \infty$.
2. $f(a + h) = \sum_{n=0}^{\infty} A_n(h, \ldots, h)$ for all $h \in X$ with $\norm{h} < r$.

The map $f$ is *analytic on $U$* if it is analytic at each point of $U$.

## Remarks

Clause 1 makes the series in clause 2 converge absolutely, since $\norm{A_n(h, \ldots, h)} \le \norm{A_n} \norm{h}^n$.
The maps $h \mapsto A_n(h, \ldots, h)$ are the continuous homogeneous polynomials of degree $n$; one may take each $A_n$ symmetric.
For $X = Y = \R$, this is the usual [[analytic function|081WKEDK6C]].
An analytic map is $C^\infty$ in the sense of the Fréchet derivative.

Example: let $B$ be a Banach algebra with unit $1$ and $\norm{xy} \le \norm{x} \norm{y}$, and let $f(x) = (1 - x)^{-1}$ for $\norm{x} < 1$.
Take $A_n(h_1, \ldots, h_n) = h_1 h_2 \cdots h_n$, so $\norm{A_n} \le 1$ for $n \ge 1$, and $r = 1$.
The Neumann series $\sum_n x^n$ converges and equals $(1 - x)^{-1}$, so $f$ is analytic at $0$.

Non-example: the [[smooth non-analytic function|09RTYYHXKA]] $f(t) = e^{-1/t^2}$, $f(0) = 0$, on $\R$ is not analytic at $0$.
All its derivatives at $0$ vanish, so any series as in clause 2 would be $0$, but $f(t) \neq 0$ for $t \neq 0$.