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A holomorphic map in infinite dimensions

Seppo I Hiltunen
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Abstract

We prove holomorphy E sqcap C(I,varPi) to C(I,varPi) of the map (x,y) mapsto x circ [id,y] where [id,y]:I owns t mapsto (t,y(t)) for a real compact interval I, and where varPi is a complex Banach space and E is a certain locally convex space of continuous functions x:I times varPi to varPi for which x(t,.) is holomorphic for all t in I. We also discuss application of this result to establishing a holomorphic solution map (xi,varphi) mapsto y for functions y:I to varPi satisfying the ordinary differential equation y' = varphi circ [id,y] with initial condition y(t_0) = xi .

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Seppo I Hiltunen. 2006-10-19. A holomorphic map in infinite dimensions. https://arxiv.org/abs/math/0609632

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