Differentiable paths in topological vector spaces
In this note, we show that a strong form of the Bolzano-Weierstrass theorem in a topological vector space E[ T] is equivalent, for example, to the assertion that there are enough differentiable paths, x (t), with non-trivial tangen t vectors, so that a function f defined on E will be sequentiall...
- Autores:
-
Findley, D. F.
- Tipo de recurso:
- Article of journal
- Fecha de publicación:
- 1974
- Institución:
- Universidad Nacional de Colombia
- Repositorio:
- Universidad Nacional de Colombia
- Idioma:
- spa
- OAI Identifier:
- oai:repositorio.unal.edu.co:unal/42385
- Acceso en línea:
- https://repositorio.unal.edu.co/handle/unal/42385
http://bdigital.unal.edu.co/32482/
- Palabra clave:
- Bolzano-Weierstrass theorem
topological vector spaces
ocally convex spaces
bounded sets
- Rights
- openAccess
- License
- Atribución-NoComercial 4.0 Internacional
| Summary: | In this note, we show that a strong form of the Bolzano-Weierstrass theorem in a topological vector space E[ T] is equivalent, for example, to the assertion that there are enough differentiable paths, x (t), with non-trivial tangen t vectors, so that a function f defined on E will be sequentially continuous for T if the composites f(x(t)) are all continuous. For a large class of locally convex spaces, this property is shown to be equivalent to the statement that the bounded sets of E[T] are finite dimensional. This leads to some very precise results for special cases. |
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