---
tag: H1KHRWNH8B
kind: concept
name: Analytic function between Banach spaces
msc: [46G20, "26E05"]
sources:
  - url: https://en.wikipedia.org/wiki/Analytic_function
---

## Summary

A map between Banach spaces is analytic at a point if near that point it is the sum of a convergent power series.
The terms of the series are continuous homogeneous polynomials, given by continuous multilinear maps.
This extends real and complex analytic functions of finitely many variables to infinite dimension.
Over the complex numbers, analytic maps are the same as holomorphic maps.

## Explanations

### L1 human

An analytic function of one variable is a convergent power series $\sum a_n x^n$.
For a map on a space of vectors, the monomial $a_n x^n$ is replaced by an expression of degree $n$ in the vector, such as $A(x, x, \ldots, x)$ for a multilinear map $A$.
A basic example is $(1 - x)^{-1} = 1 + x + x^2 + \cdots$ for a square matrix $x$ of small norm.