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The convenient setting for real analytic mappings. (English) Zbl 0738.46024

In this article, we present a careful study of real analytic mappings in infinite (and finite) dimensions combined with a thorough treatment of locally convex topologies on spaces of real analytic functions. From the beginning our aim is Cartesian closedness: a mapping \(f: E\times F\to G\) should be real analytic if and only if the canonically associated mapping \(\check f: E\to C^ \omega(F, G)\) is it. Very simple examples, see 1.1, show that real analytic in the sense of having a locally converging Taylor series is too restrictive.
The right notion turns out to be scalarwise real analytic: A curve in a locally convex space is called (scalarwise) real analytic if and only if composed with each continuous linear functional it gives a real analytic function. Later we show, that the space of real analytic curves does not depend on the topology, only on the bornology described by the dual.
A mapping will be called real analytic if it maps smooth curves to smooth curves and real analytic curves to real analytic curves. This definition is in spirit very near to the original ideas of variational calculus and it leads to a simple and powerful theory. We will show the surprising result, that under some mild completeness conditions (i.e. for convenient vector spaces), the second condition can be replaced by: the mapping should be real analytic along affine lines, see 2.7. This is a version of Hartogs’ theorem, which for Banach spaces is due to J. Bochnak and J. Siciak.
It is a very satisfying result, that the right realm of spaces of real analytic analysis is the category of convenient vector spaces, which is also the good setting in infinite dimensions for smooth analysis, see [A. Fröhlicher, A. Kriegl, Linear spaces and differentiation theory (1988; Zbl 0657.46034)], and for holomorphic analysis, see [A. Kriegl, L. D. Nel, Cah. Top. Geom. Differ. Cat. 26, 273-309 (1985; Zbl 0581.46041)].
The later parts of this paper are devoted to the study of manifolds of real analytic mappings. We show indeed, that the set of real analytic mappings from a compact manifold to another one is a real analytic manifold, that composition is real analytic and that the group of real analytic diffeomorphisms is a real analytic Lie group. The exponential mapping of it (integration of vector fields) is real analytic, but as in the smooth case it is still not surjective on any neighborhood of the identity. We would like to stress the fact that the group of smooth diffeomorphisms of a manifold is a smooth but not a real analytic Lie group. We also show that the space of smooth mappings between real analytic manifolds is a real analytic manifold, but the composition is only smooth.
Throughout this paper our basic guiding line is the Cartesian closed calculus for smooth mappings as exposed in [Fröhlicher, Kriegl, op. cit.]. The reader is assumed to be familiar with at least the rudiments of it; but section 1 contains a short summary of the essential parts.

MSC:

46G20 Infinite-dimensional holomorphy
46E50 Spaces of differentiable or holomorphic functions on infinite-dimensional spaces
58B12 Questions of holomorphy and infinite-dimensional manifolds
46A17 Bornologies and related structures; Mackey convergence, etc.
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