A unified approach to Banach-Stone theorem on spaces of differentiable functions under various norms
The classical Banach-Stone Theorem asserts that the isometric structure of the space of real-valued continuous functions determines a compact Hausdorf space up to homeomorphism. There are various extensions and generalizations of the result in different contexts. One of them is to consider vector...
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| Format: | Thesis-Doctor of Philosophy |
| Language: | English |
| Published: |
Nanyang Technological University
2021
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| Online Access: | https://hdl.handle.net/10356/148685 |
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| Institution: | Nanyang Technological University |
| Language: | English |
| Summary: | The classical Banach-Stone Theorem asserts that the isometric structure of the
space of real-valued continuous functions determines a compact Hausdorf space
up to homeomorphism. There are various extensions and generalizations of the
result in different contexts. One of them is to consider vector-valued differentiable
function space endowed with different norms. Existing literatures utilized special
geometrical properties of norms to obtain a variant of the result. However, their
methods are restricted to the considered norms only. In this thesis, we developed
a general framework, which provides a sufficient condition to obtain Banach-
Stone Theorem for vector-valued differentiable function space. Then we applied
the framework on two different norms, which generalized most norms considered
in existing literatures. When restricting them to \ell^p-norms, where p in [1,infinity), we
obtained a characterization of Banach-Stone Theorem. |
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