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1 - Calculus in Locally Convex Spaces

Published online by Cambridge University Press:  08 December 2022

Alexander Schmeding
Affiliation:
Nord Universitet, Norway
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Summary

It is well known that multidimensional calculus, aka Fréchet calculus, carries over to the realm of Banach spaces and Banach manifolds. Banach spaces are often not sufficient for our purposes. To generalise derivatives, we will, as a minimum, need vector spaces with an amenable topology (which need not be induced by a norm). This chapter presents first a notion of calculus in locally convex spaces, which requires the existence and continuity of directional derivatives. The resulting calculus is called Bastiani calculus and we compare it to some common (but inequivalent) notions of calculus such as the convenient calculus. Building on the chain rule, we then construct the basic building blocks of (infinite-dimensional) differential geometry: manifolds and their tangent spaces as well as submersions and immersions.

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Publisher: Cambridge University Press
Print publication year: 2022
Creative Commons
Creative Common License - CCCreative Common License - BYCreative Common License - NCCreative Common License - ND
This content is Open Access and distributed under the terms of the Creative Commons Attribution licence CC-BY-NC-ND 4.0 https://creativecommons.org/cclicenses/

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