# Mappings between Banach spaces

What is the definition of an analytic mapping between two Banach spaces? This is a problem I ran into when solving an integral equation. One of the related coefficients is represented as a functional $A(\xi, u(x)), \xi\in\mathbb{R}$ of a function $u(x)\in L^1(\mathbb{R})\cap L^2(\mathbb{R})$. I have completed an estimation of the $L^1(\mathbb{R})$ and $L^2(\mathbb{R})$ norm of $A$, which are both bounded by $e^{||u||_{L^1}}$$||u||_{L^2}$. Now I want to find out the dependence of $A$ on the function $u$. I know how to define continuity in this case, but how is one supposed to define the differentiable and analytic mapping here? Is there any standard reference on this stuff? Thank you!

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MR0842435 (88d:46084) Reviewed Mujica, Jorge(BR-ECP) Complex analysis in Banach spaces. Holomorphic functions and domains of holomorphy in finite and infinite dimensions. North-Holland Mathematics Studies, 120. Notas de Matemática [Mathematical Notes], 107. North-Holland Publishing Co., Amsterdam, 1986. xii+434 pp. ISBN: 0-444-87886-6 46G20 (32Dxx 32Exx 58C10) – Bill Johnson Jun 20 '13 at 6:06
Dineen, Seán Complex analysis on infinite-dimensional spaces. Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 1999. xvi+543 pp. ISBN: 1-85233-158-5 (Reviewer: José Bonet) 46G20 (32Kxx 46A04 46E50 46G25 58B12) A simple Google search does fine here, so I vote to close. – Bill Johnson Jun 20 '13 at 6:08
Chapter II of: Andreas Kriegl, Peter W. Michor: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, Volume: 53, American Mathematical Society, Providence, 1997, mat.univie.ac.at/~michor/apbookh-ams.pdf – Peter Michor Jun 20 '13 at 9:05