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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