## Abstract

We start the systematic study of Fréchet spaces which are ℵ-spaces in the weak topology. A topological space X is an ℵ_{0}-space or an ℵ-space if X has a countable k-network or a σ-locally finite k-network, respectively. We are motivated by the following result of Corson (1966): If the space C_{c}(X) of continuous real-valued functions on a Tychonoff space X endowed with the compact-open topology is a Banach space, then C_{c}(X) endowed with the weak topology is an ℵ_{0}-space if and only if X is countable. We extend Corson's result as follows: If the space E:=C_{c}(X) is a Fréchet lcs, then E endowed with its weak topology σ(E, E^{'}) is an ℵ-space if and only if (E, σ(E, E^{'})) is an ℵ_{0}-space if and only if X is countable. We obtain a necessary and some sufficient conditions on a Fréchet lcs to be an ℵ-space in the weak topology. We prove that a reflexive Fréchet lcs E in the weak topology σ(E, E^{'}) is an ℵ-space if and only if (E, σ(E, E^{'})) is an ℵ_{0}-space if and only if E is separable. We show however that the nonseparable Banach space ℓ1(R) with the weak topology is an ℵ-space.

Original language | English |
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Pages (from-to) | 1183-1199 |

Number of pages | 17 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 432 |

Issue number | 2 |

DOIs | |

State | Published - 15 Dec 2015 |

## Keywords

- Fréchet space
- Space of continuous functions
- Weakly ℵ locally convex space
- ℵ-space
- ℵ-space