# Banach spaces with the (strong) Gelfand--Phillips property

@inproceedings{Banakh2021BanachSW, title={Banach spaces with the (strong) Gelfand--Phillips property}, author={Taras O. Banakh and Saak Gabriyelyan}, year={2021} }

Several new characterizations of the Gelfand–Phillips property are given. We define a strong version of the Gelfand–Phillips property and prove that a Banach space has this stronger property iff it embeds into c0. For an infinite compact space K, the Banach space C(K) has the strong Gelfand–Phillips property iff C(K) is isomorphic to c0 iff K is countable and has finite scattered height.

## 3 Citations

Locally convex spaces with the strong Gelfand-Phillips property

- Mathematics
- 2021

We introduce the strong Gelfand–Phillips property for locally convex spaces and give several characterizations of this property. We characterize the strong Gelfand–Phillips property among locally…

The Gelfand-Phillips property for locally convex spaces

- Mathematics
- 2021

We extend the well-known Gelfand–Phillips property for Banach spaces to locally convex spaces, defining a locally convex space E to be Gelfand–Phillips if every limited set in E is precompact in the…

A simple Efimov space with sequentially-nice space of probability measures

- Mathematics
- 2021

Under Jensen’s diamond principle ♦, we construct a simple Efimov space K whose space of nonatomic probability measures Pna(K) is first-countable and sequentially compact. These two properties of…

## References

SHOWING 1-10 OF 40 REFERENCES

On Banach spaces with the Gelfand-Phillips property. II

- Mathematics
- 1989

We present some result of lifting of the Gelfand Phillips property from Banach spacesE andF to Banach spaces of compact operators and of Bochner integrable functions. Moreover we studyC(K) spaces…

On the Gelfand-Phillips property in Banach spaces with PRI

- Mathematics
- 1997

It is proved that every Banach space belonging to a certain class called the class $\mathcal{P}$ possesses the Gelfand-Phillips property. Consequently, so does every weakly countably determined…

THE GELFAND–PHILLIPS PROPERTY IN CLOSED SUBSPACES OF SOME OPERATOR SPACES

- Mathematics
- 2011

By introducing the concept of limited completely continuous op- erators between two arbitrary Banach spaces X and Y , we give some properties of this concept related to some well known classes of…

A Gelfand‐Phillips Property with Respect to the Weak Topology

- Mathematics
- 1990

We consider a Gelfand-Phillips type property for the weak topology. The main results that we obtain are (1) for certain Banach spaces, E˜ϵF inherits this property from E and F, and (2) the spaces…

Limited Sets in C(K)‐Spaces and Examples Concerning the Gelfand‐Phillips‐Property

- Mathematics
- 1992

In this paper we give criteria for limitedness in C(K)-spaces and discuss the Gelfand-Phillips-property. We show that the Gelfand-Phillips-property is not a three-space-property, that l1 ⊄ X does not…

Banach Space Theory: The Basis for Linear and Nonlinear Analysis

- Mathematics
- 2010

Preface.- Basic Concepts in Banach Spaces.- Hahn-Banach and Banach Open Mapping Theorems.- Weak Topologies and Banach Spaces.- Schauder Bases.- Structure of Banach Spaces.- Finite-Dimensional…

A simple Efimov space with sequentially-nice space of probability measures

- Mathematics
- 2021

Under Jensen’s diamond principle ♦, we construct a simple Efimov space K whose space of nonatomic probability measures Pna(K) is first-countable and sequentially compact. These two properties of…

A Gelfand-Phillips space not containing l1 whose dual ball is not weak * sequentially compact

- MathematicsGlasgow Mathematical Journal
- 2001

A set D in a Banach space E is called limited if pointwise convergent sequences of linear functionals converge uniformly on D and E is called a GP-space (after Gelfand and Phillips) if every limited…