arXiv subjects

Helge Glockner

Publications and source records attributed to Helge Glockner.

75 records · Page 5Linked to original sources

arXiv2003

Examples of differentiable mappings into non-locally convex spaces

Examples of differentiable mappings into real or complex topological vector spaces with specific properties are given, which illustrate the differences between differential calculus in the locally convex and the non-locally convex case. In particular, for a suitable non-locally convex space E, we describe a smooth injection of R into E whose derivative vanishes identically; we present a complex C^\infty-map on the complex field C which is not given locally by its Taylor series, around any point; we present a complex C^1-map into a complete, non-locally convex topological vector space which is not C^2; and we present a compactly supported, non-zero, complex C^\infty-map from C to a suitable non-locally convex space.

math.FA↗

arXiv2003

Differential Calculus, Manifolds and Lie Groups over Arbitrary Infinite Fields

We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of manifolds and Lie groups is developed. Special attention is paid to the case of mappings between topological vector spaces over non-discrete topological fields, in particular ultrametric fields or the fields of real and complex numbers. In the latter case, a theory of differentiable mappings between general, not necessarily locally convex spaces is obtained, which in the locally convex case is equivalent to Keller's C^k_c-theory.

math.GM↗

arXiv2003

Implicit Functions from Topological Vector Spaces to Banach Spaces

We prove implicit function theorems for mappings on topological vector spaces over valued fields. In the real and complex cases, we obtain implicit function theorems for mappings from arbitrary (not necessarily locally convex) topological vector spaces to Banach spaces.

math.GM↗