TY - GEN
T1 - Topological Properties of the Weak and Weak ∗ Topologies of Function Spaces
AU - Gabriyelyan, Saak
N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.
PY - 2023/1/1
Y1 - 2023/1/1
N2 - We show that if a locally convex space (lcs) (E, τ) is a sequentially Ascoli space in the weak topology, then τ= σ(E, E′). Consequently, a complete lcs E is weakly sequentially Ascoli iff E= Fλ for some cardinal λ. For a Tychonoff space X, let Ck(X) be the space C(X) of all continuous functions on X endowed with the compact-open topology. We prove that: (1) Ck(X) is weakly sequentially Ascoli iff Ck(X) is weakly Ascoli iff it is weakly κ -Fréchet–Urysohn iff X has the property (κ) and every compact subset of X is finite; (2) Ck(X) is weakly Fréchet–Urysohn iff Ck(X) is a weakly sequential space iff it is a weakly k-space iff the space X has the property γ whose compact subsets are finite; (3) the dual space of Ck(X) is weak ∗ sequentially Ascoli iff X is finite. If Cb(X) is the space C(X) with the topology of uniform convergence on functionally bounded subsets of X, then: (a) Cb(X) is weakly sequentially Ascoli iff Cb(X) is weakly Ascoli iff it is weakly κ -Fréchet–Urysohn iff X has the property (κ) and every functionally bounded subset of X is finite; (2) Cb(X) is weakly Fréchet–Urysohn iff Cb(X) is a weakly sequential space iff it is a weakly k-space iff the space X has the property γ whose functionally bounded subsets are finite.
AB - We show that if a locally convex space (lcs) (E, τ) is a sequentially Ascoli space in the weak topology, then τ= σ(E, E′). Consequently, a complete lcs E is weakly sequentially Ascoli iff E= Fλ for some cardinal λ. For a Tychonoff space X, let Ck(X) be the space C(X) of all continuous functions on X endowed with the compact-open topology. We prove that: (1) Ck(X) is weakly sequentially Ascoli iff Ck(X) is weakly Ascoli iff it is weakly κ -Fréchet–Urysohn iff X has the property (κ) and every compact subset of X is finite; (2) Ck(X) is weakly Fréchet–Urysohn iff Ck(X) is a weakly sequential space iff it is a weakly k-space iff the space X has the property γ whose compact subsets are finite; (3) the dual space of Ck(X) is weak ∗ sequentially Ascoli iff X is finite. If Cb(X) is the space C(X) with the topology of uniform convergence on functionally bounded subsets of X, then: (a) Cb(X) is weakly sequentially Ascoli iff Cb(X) is weakly Ascoli iff it is weakly κ -Fréchet–Urysohn iff X has the property (κ) and every functionally bounded subset of X is finite; (2) Cb(X) is weakly Fréchet–Urysohn iff Cb(X) is a weakly sequential space iff it is a weakly k-space iff the space X has the property γ whose functionally bounded subsets are finite.
KW - Function space
KW - Sequential space
KW - Sequentially Ascoli space
KW - Weak topology
KW - Weak topology
UR - http://www.scopus.com/inward/record.url?scp=85169073147&partnerID=8YFLogxK
U2 - 10.1007/978-3-031-30014-1_6
DO - 10.1007/978-3-031-30014-1_6
M3 - Conference contribution
AN - SCOPUS:85169073147
SN - 9783031300134
T3 - Springer Proceedings in Mathematics and Statistics
SP - 123
EP - 135
BT - Functional Analysis and Continuous Optimization - In Honour of Juan Carlos Ferrando’s 65th Birthday, 2022
A2 - Amigó, José M.
A2 - Cánovas, María J.
A2 - López-Cerdá, Marco A.
A2 - López-Pellicer, Manuel
PB - Springer
T2 - International Meeting on Functional Analysis and Continuous Optimization, IMFACO 2022
Y2 - 16 June 2022 through 17 June 2022
ER -