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  "body": "\n## Summary\n\nA map between Banach spaces is analytic at a point if near that point it is the sum of a convergent power series.\nThe terms of the series are continuous homogeneous polynomials, given by continuous multilinear maps.\nThis extends real and complex analytic functions of finitely many variables to infinite dimension.\nOver the complex numbers, analytic maps are the same as holomorphic maps.\n\n## Explanations\n\n### L1 human\n\nAn analytic function of one variable is a convergent power series $\\sum a_n x^n$.\nFor 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 $A$.\nA basic example is $(1 - x)^{-1} = 1 + x + x^2 + \\cdots$ for a square matrix $x$ of small norm.\n",
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