On weakly Lindelöf Banach spaces.

*(English)*Zbl 0797.46009Summary: We define and investigate the properties of a proper class of Banach spaces each member of which is Lindelöf in its weak topology; we call them weakly Lindelöf determined (WLD) Banach spaces. The class of WLD Banach spaces extends the known class of WCD (weakly countably determined) Banach spaces and inherits some of its basic properties: (e.g., each WLD Banach space has a projectional resolution of identity, and it is also derived from a small weakly Lindelöf subset, etc.).

We also present several examples, in our attempt to clarify the concept of weakly countably determinedness, such as:

(i) a WLD Banach space which is dually strictly convexifiable, but not WCD;

(ii) a WLD Banach space with an unconditional basis, which is not weak Asplund, whose dual space is strictly convexifiable;

(iii) a dual weakly \(K\)-analytic Banach space which is not a subspace of a weakly compactly generated Banach space.

On the grounds of these examples, we answer questions and problems of Gruenhage, Larman and Phelps, and Talagrand.

We also present several examples, in our attempt to clarify the concept of weakly countably determinedness, such as:

(i) a WLD Banach space which is dually strictly convexifiable, but not WCD;

(ii) a WLD Banach space with an unconditional basis, which is not weak Asplund, whose dual space is strictly convexifiable;

(iii) a dual weakly \(K\)-analytic Banach space which is not a subspace of a weakly compactly generated Banach space.

On the grounds of these examples, we answer questions and problems of Gruenhage, Larman and Phelps, and Talagrand.

##### MSC:

46B10 | Duality and reflexivity in normed linear and Banach spaces |

46B25 | Classical Banach spaces in the general theory |

46A50 | Compactness in topological linear spaces; angelic spaces, etc. |

##### Keywords:

Corson-compact spaces; weakly Lindelöf determined Banach spaces; weakly countably determined Banach spaces; dual weakly \(K\)-analytic Banach space; Lindelöf in its weak topology; WLD Banach spaces; weakly countably determinedness; WCD
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\textit{S. Argyros} and \textit{S. Mercourakis}, Rocky Mt. J. Math. 23, No. 2, 395--446 (1993; Zbl 0797.46009)

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