Optimal average approximations for functions mapping in quasi-Banach spaces
In 1994, M.M. Popov [6] showed that the fundamental theorem of calculus fails, in general, for functions mapping from a compact interval of the real line into the ℓp-spaces for 0<p<1, and the question arose whether such a significant result might hold in some other non-Banach spaces. I...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2014 |
| País: | España |
| Institución: | Universidad de La Rioja (UR) |
| Repositorio: | RIUR. Repositorio Institucional de la Universidad de La Rioja |
| OAI Identifier: | oai:portal.dialnet.es:doc/5bbc6a08b750603269e825d6 |
| Acceso en línea: | https://investigacion.unirioja.es/documentos/5bbc6a08b750603269e825d6 |
| Access Level: | acceso abierto |
| Palabra clave: | Fundamental theorem of calculus Optimal approximation Quasi-Banach space Riemann integral |
| Sumario: | In 1994, M.M. Popov [6] showed that the fundamental theorem of calculus fails, in general, for functions mapping from a compact interval of the real line into the ℓp-spaces for 0<p<1, and the question arose whether such a significant result might hold in some other non-Banach spaces. In this article we completely settle the problem by proving that the fundamental theorem of calculus breaks down in the context of any non-locally convex quasi-Banach space. Our approach introduces the tool of Riemann-integral averages of continuous functions, and uses it to bring out to light the differences in behavior of their approximates in the lack of local convexity. As a by-product of our work we solve a problem raised in [1] on the different types of spaces of differentiable functions with values on a quasi-Banach space. © 2013 Elsevier Inc. |
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