---
tag: 2VQDK71A27
kind: statement
role: theorem
name: Namioka--Phelps
msc: [46-XX, 54-XX]
sources:
  - arxiv: 2510.01873v1
    locator: Theorem 2.5
    mention: ax://paper/2510.01873#5CZ6
    licence: cc-by
    note: the paper that states the item (corpus arxiv AX9.2)
history:
  notes: The source is arXiv:2510.01873v1 (2025-10), by Marian Fabian, Jerzy Kcakol, Arkady Leiderman. A paper also restates older results, so this is not the first statement until a person or a later pass confirms it.
inventory: [PQM04SEB51]
uses: [0QPHW703PV, D38YWGXRKJ, 06GZ956657, 01PKQYTTMT, JKQP7AFZ0E, 4A82S5FVRT]
---

## Statement

Let $X$ be a [[compact space|0QPHW703PV]]. Let $C(X)$ be the [[space of bounded continuous functions|D38YWGXRKJ]] on $X$, which is a [[Banach space|06GZ956657]] under the supremum norm.

A Banach space $E$ is *Asplund* when each closed separable subspace of $E$ has a separable [[continuous dual space|4A82S5FVRT]].

The space $X$ is *scattered* when each nonempty subset $A \subseteq X$ has a point that is [[isolated|01PKQYTTMT]] in $A$ for the subspace topology.

Then $C(X)$ is Asplund if and only if $X$ is scattered.