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1 results

  • 1 - Calculus in Locally Convex Spaces

  • Alexander Schmeding, Nord Universitet, Norway
  • Book:
    An Introduction to Infinite-Dimensional Differential Geometry
    Published online:
    08 December 2022
    Print publication:
    22 December 2022, pp 1-29
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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.