Namioka--Phelps
Builds on Compact spaceSpace of bounded continuous functionsBanach spaceIsolated pointSeparable Banach spaceContinuous dual space
Statement
Let be a compact space. Let be the space of bounded continuous functions on , which is a Banach space under the supremum norm.
A Banach space is Asplund when each closed separable subspace of has a separable continuous dual space.
The space is scattered when each nonempty subset has a point that is isolated in for the subspace topology.
Then is Asplund if and only if is scattered.
Reading path
To understand this page from the ground up, read these 41 pages in order. The foundations come first.
- Banach space concept
- Continuous dual space concept
- Continuous function concept
- Dense set concept
- Element definition
- Normed vector space concept
- Proposition concept
- Real numbers object
- Separable space concept
- Set bounded above definition
- Topological space concept
- Separable Banach space definition
- Set definition
- Conjunction definition
- Disjunction definition
- Implication definition
- Predicate definition
- Supremum definition
- Empty set object
- Family of sets definition
- Ordered pair definition
- Intersection definition
- Union definition
- Existential quantifier definition
- Universal quantifier definition
- Space of bounded continuous functions definition
- Cartesian product definition
- Union of a family of sets definition
- Subset definition
- Uniqueness quantifier definition
- Binary relation definition
- Topological space definition
- Function definition
- Metric space definition
- Open ball definition
- Metric topology definition
- Open set definition
- Isolated point definition
- Open cover definition
- Compact space definition
- Namioka--Phelps statement
Further reading
- mention ax://paper/2510.01873#5CZ6, Theorem 2.5. the paper that states the item (corpus arxiv AX9.2)
In the catalogue
- Asplund spaces $C_k(X)$ beyond Banach spaces, Marian Fabian, Jerzy Kcakol, Arkady Leiderman
About this page
| Tag | 2VQDK71A27 |
|---|---|
| URI | m2://2VQDK71A27 |
| Kind | statement, theorem |
| Writing | W1 stated |
| MSC | 46-XX, 54-XX |
| Status | This copy of the site has no build data. Run m2 status 2VQDK71A27. |
| Uses | Compact space, Space of bounded continuous functions, Banach space, Isolated point, Separable Banach space, Continuous dual space |
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What this page needs for W2
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- edge proves in: at least one proof node (0 of 1)
- edge instance_of or illustrates in: one example that shows the statement at work (0 of 1)
- explanation L2 (0 of 1)
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History
Each change to this page, from its provenance log. A change by an agent names its run.
| Date | Change | Field | By |
|---|---|---|---|
| 2026-11-02 | add | the whole node | tamnd |
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