L-functional analysis
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arXiv
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| Main Authors: | , , , , , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929681205297152 |
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| author | Kikianty, Eder Messerschmidt, Miek Naude, Luan Roelands, Mark Schwanke, Christopher van Amstel, Walt van der Walt, Jan Harm Wortel, Marten |
| author_facet | Kikianty, Eder Messerschmidt, Miek Naude, Luan Roelands, Mark Schwanke, Christopher van Amstel, Walt van der Walt, Jan Harm Wortel, Marten |
| contents | Inspired by the theories of Kaplansky-Hilbert modules and probability theory in vector lattices, we generalise functional analysis by replacing the scalars $\mathbb{R}$ or $\mathbb{C}$ by a real or complex Dedekind complete unital $f$-algebra $\mathbb{L}$; such an algebra can be represented as a suitable space of continuous functions. We set up the basic theory of $\mathbb{L}$-normed and $\mathbb{L}$-Banach spaces and bounded operators between them, we discuss the $\mathbb{L}$-valued analogues of the classical $\ell^p$-spaces, and we prove the analogue of the Hahn-Banach theorem. We also discuss the basics of the theory of $\mathbb{L}$-Hilbert spaces, including projections onto convex subsets, the Riesz Representation theorem, and representing $\mathbb{L}$-Hilbert spaces as a direct sum of $\ell^2$-spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_10222 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | L-functional analysis Kikianty, Eder Messerschmidt, Miek Naude, Luan Roelands, Mark Schwanke, Christopher van Amstel, Walt van der Walt, Jan Harm Wortel, Marten Functional Analysis 46B99 (Primary), 06F25 (Secondary) Inspired by the theories of Kaplansky-Hilbert modules and probability theory in vector lattices, we generalise functional analysis by replacing the scalars $\mathbb{R}$ or $\mathbb{C}$ by a real or complex Dedekind complete unital $f$-algebra $\mathbb{L}$; such an algebra can be represented as a suitable space of continuous functions. We set up the basic theory of $\mathbb{L}$-normed and $\mathbb{L}$-Banach spaces and bounded operators between them, we discuss the $\mathbb{L}$-valued analogues of the classical $\ell^p$-spaces, and we prove the analogue of the Hahn-Banach theorem. We also discuss the basics of the theory of $\mathbb{L}$-Hilbert spaces, including projections onto convex subsets, the Riesz Representation theorem, and representing $\mathbb{L}$-Hilbert spaces as a direct sum of $\ell^2$-spaces. |
| title | L-functional analysis |
| topic | Functional Analysis 46B99 (Primary), 06F25 (Secondary) |
| url | https://arxiv.org/abs/2403.10222 |