How to describe the topology of cosmic.... ℵ 0 -spaces? Endow N N with the order, i.e., α ≤ β if α i ≤ β i for all i ∈ N , α = ( α i ) i ∈ N , β = ( β i ) i ∈ N . For every α ∈ N N , k ∈ N , set β ∈ N N : β i = α i for i = 1 , . . . , k � � I k ( α ) := . Let M ⊆ N N and U = { U α : α ∈ M } be an M -decreasing family of subsets of a set X . Define the countable family D U of subsets of X by � D U := { D k ( α ) : α ∈ M , k ∈ N } , where D k ( α ) := U β , β ∈ I k ( α ) ∩ M U satisfies condition ( D ) if U α = � k ∈ N D k ( α ), α ∈ M . ( X , τ ) has a small base if there exists an M -decreasing base of τ for some M ⊆ N N [Gabriyelyan-K.]. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Theorem 5 (Gabriyelyan-K.) (i) X is cosmic iff X has a small base U = { U α : α ∈ M } with condition ( D ) . In that case the family D U is a countable network in X. (ii) X is an ℵ 0 -space iff X has a small base U = { U α : α ∈ M } with condition ( D ) such that the family D U is a countable k-network in X. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Theorem 5 (Gabriyelyan-K.) (i) X is cosmic iff X has a small base U = { U α : α ∈ M } with condition ( D ) . In that case the family D U is a countable network in X. (ii) X is an ℵ 0 -space iff X has a small base U = { U α : α ∈ M } with condition ( D ) such that the family D U is a countable k-network in X. Corollary 6 Let G be a Baire topological group. Then G is cosmic iff G is metrizable and separable. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
When B w is an ℵ - and k -space? JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
When B w is an ℵ - and k -space? The following classical fact will be used later: Theorem 7 (Schl¨ uchtermann-Wheeler) The following are equivalent for a Banach space E. (i) B w is Fr´ echet–Urysohn. (ii) B w is sequential. (iii) B w is a k-space, i.e. P ⊂ B w is closed in B w if P ∩ K is closed in K for all compact K ⊂ B w . (iv) E contains no isomorphic copy of ℓ 1 . JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
When B w is an ℵ - and k -space? The following classical fact will be used later: Theorem 7 (Schl¨ uchtermann-Wheeler) The following are equivalent for a Banach space E. (i) B w is Fr´ echet–Urysohn. (ii) B w is sequential. (iii) B w is a k-space, i.e. P ⊂ B w is closed in B w if P ∩ K is closed in K for all compact K ⊂ B w . (iv) E contains no isomorphic copy of ℓ 1 . Theorem 8 (Schl¨ uchtermann-Wheeler) If E is a Banach space, then E w is a k-space iff dim( E ) < ∞ . JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Theorem 9 (Schl¨ uchtermann-Wheeler) The following conditions are equivalent for a Banach space E. (i) B w is (separable) metrizable. (ii) B w is an ℵ 0 -space and a k-space. (iii) The dual E ′ is separable. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Theorem 9 (Schl¨ uchtermann-Wheeler) The following conditions are equivalent for a Banach space E. (i) B w is (separable) metrizable. (ii) B w is an ℵ 0 -space and a k-space. (iii) The dual E ′ is separable. Theorem 10 (Gabriyelyan-K.-Zdomskyy) The following conditions on a Banach space E are equivalent: (i) B w is (separable) metrizable. (ii) B w is an ℵ -space and a k-space. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Banach spaces for which E w is an ℵ -space. Hence, the assumption on C ( K ) w to have a σ -locally finite k -network is much to strong. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Banach spaces for which E w is an ℵ -space. Problem 11 Describe those Banach spaces E for which E w is an ℵ -space. Hence, the assumption on C ( K ) w to have a σ -locally finite k -network is much to strong. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Banach spaces for which E w is an ℵ -space. Problem 11 Describe those Banach spaces E for which E w is an ℵ -space. Theorem 12 (Corson) C [0 , 1] w is not an ℵ 0 -space. Hence, the assumption on C ( K ) w to have a σ -locally finite k -network is much to strong. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Banach spaces for which E w is an ℵ -space. Problem 11 Describe those Banach spaces E for which E w is an ℵ -space. Theorem 12 (Corson) C [0 , 1] w is not an ℵ 0 -space. Theorem 13 (Gabriyelyan-K.-Kubi´ s-Marciszewski) For a Banach space E := C ( K ) the space E w is an ℵ -space iff E w is an ℵ 0 -space iff K is countable. Hence, the assumption on C ( K ) w to have a σ -locally finite k -network is much to strong. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Theorem 14 (Gabriyelyan-K.-Kubi´ s-Marciszewski) Let E be a Banach space not containing a copy of ℓ 1 . The following conditions are equivalent: (i) E w is an ℵ -space (ii) E w is an ℵ 0 -space. (iii) The dual E ′ is separable. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Theorem 14 (Gabriyelyan-K.-Kubi´ s-Marciszewski) Let E be a Banach space not containing a copy of ℓ 1 . The following conditions are equivalent: (i) E w is an ℵ -space (ii) E w is an ℵ 0 -space. (iii) The dual E ′ is separable. ( ℓ 1 ) w is an ℵ 0 -space. ( JT ) w is a σ -space but not an ℵ -space. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Theorem 14 (Gabriyelyan-K.-Kubi´ s-Marciszewski) Let E be a Banach space not containing a copy of ℓ 1 . The following conditions are equivalent: (i) E w is an ℵ -space (ii) E w is an ℵ 0 -space. (iii) The dual E ′ is separable. ( ℓ 1 ) w is an ℵ 0 -space. ( JT ) w is a σ -space but not an ℵ -space. Corollary 15 If E is separable and does not contain ℓ 1 , then E w is an ℵ 0 -space iff E ′ has a w ∗ -Kadec norm. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Theorem 16 (Gabriyelyan-K.-Kubi´ s-Marciszewski) ( ℓ 1 (Γ)) w is an ℵ -space iff the cardinality of Γ does not exceed the continuum. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Theorem 16 (Gabriyelyan-K.-Kubi´ s-Marciszewski) ( ℓ 1 (Γ)) w is an ℵ -space iff the cardinality of Γ does not exceed the continuum. ℓ 1 (Γ) with the weak topology does not have countable pseudocharacter whenever | Γ | > 2 ℵ 0 . JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Theorem 16 (Gabriyelyan-K.-Kubi´ s-Marciszewski) ( ℓ 1 (Γ)) w is an ℵ -space iff the cardinality of Γ does not exceed the continuum. ℓ 1 (Γ) with the weak topology does not have countable pseudocharacter whenever | Γ | > 2 ℵ 0 . Hence ( ℓ 1 ( R )) w is an ℵ -space which is not an ℵ 0 -space and ( ℓ 1 ( R )) w is not normal. Last claim follows from: JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Theorem 16 (Gabriyelyan-K.-Kubi´ s-Marciszewski) ( ℓ 1 (Γ)) w is an ℵ -space iff the cardinality of Γ does not exceed the continuum. ℓ 1 (Γ) with the weak topology does not have countable pseudocharacter whenever | Γ | > 2 ℵ 0 . Hence ( ℓ 1 ( R )) w is an ℵ -space which is not an ℵ 0 -space and ( ℓ 1 ( R )) w is not normal. Last claim follows from: Theorem 17 (Reznichenko) Let E be a Banach space. Then E w is Lindel¨ of iff E w is normal iff E w is paracompact. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
When C ( K ) w is a σ -space? JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
When C ( K ) w is a σ -space? ℵ -spaces C p ( X ) and C ( K ) w are already characterized. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
When C ( K ) w is a σ -space? ℵ -spaces C p ( X ) and C ( K ) w are already characterized. Any σ -space is perfect [Gruenhage], so σ -spaces have countable pseudocharacter. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
When C ( K ) w is a σ -space? ℵ -spaces C p ( X ) and C ( K ) w are already characterized. Any σ -space is perfect [Gruenhage], so σ -spaces have countable pseudocharacter. If E w is a σ -space, then E ′ has weak ∗ -dual separable but ( ℓ ∞ ) w is not a σ -space although ℓ ∞ has weak ∗ -dual separable. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
When C ( K ) w is a σ -space? ℵ -spaces C p ( X ) and C ( K ) w are already characterized. Any σ -space is perfect [Gruenhage], so σ -spaces have countable pseudocharacter. If E w is a σ -space, then E ′ has weak ∗ -dual separable but ( ℓ ∞ ) w is not a σ -space although ℓ ∞ has weak ∗ -dual separable. Example 18 Let Γ be an infinite set and E := ℓ p (Γ) with 1 < p < ∞ . Then ψ ( E w ) ≥ | Γ | , where E w := ( E , σ ( E , E ′ )). Hence ℓ p (Γ) w are not σ -spaces for any uncountable Γ. More: E w for any nonseparable weakly Lindel¨ of E is not a σ -space. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
When C ( K ) w is a σ -space? ℵ -spaces C p ( X ) and C ( K ) w are already characterized. Any σ -space is perfect [Gruenhage], so σ -spaces have countable pseudocharacter. If E w is a σ -space, then E ′ has weak ∗ -dual separable but ( ℓ ∞ ) w is not a σ -space although ℓ ∞ has weak ∗ -dual separable. Example 18 Let Γ be an infinite set and E := ℓ p (Γ) with 1 < p < ∞ . Then ψ ( E w ) ≥ | Γ | , where E w := ( E , σ ( E , E ′ )). Hence ℓ p (Γ) w are not σ -spaces for any uncountable Γ. More: E w for any nonseparable weakly Lindel¨ of E is not a σ -space. How to describe σ -spaces C ( K ) w ? Let’s recall the concept of descriptive Banach spaces . JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Theorem 19 (M-O-T-V-Hansell) E is descriptive [Hansell] (i.e. E has a norm-network which is σ -isolated in E w ) iff E has the JNR-property iff E w has a σ -isolated network. E has JNR iff for any ǫ > 0 there is a sequence ( E ǫ n ) covering E such that for any n ∈ N and any x ∈ E ǫ n there is an w -open neighbourhood x ∈ U with diam ( U ∩ E ǫ n ) < ǫ . JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Theorem 19 (M-O-T-V-Hansell) E is descriptive [Hansell] (i.e. E has a norm-network which is σ -isolated in E w ) iff E has the JNR-property iff E w has a σ -isolated network. E has JNR iff for any ǫ > 0 there is a sequence ( E ǫ n ) covering E such that for any n ∈ N and any x ∈ E ǫ n there is an w -open neighbourhood x ∈ U with diam ( U ∩ E ǫ n ) < ǫ . WCG ⇒ LUR ⇒ Kadec ⇒ JNR ⇔ descriptive. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Theorem 19 (M-O-T-V-Hansell) E is descriptive [Hansell] (i.e. E has a norm-network which is σ -isolated in E w ) iff E has the JNR-property iff E w has a σ -isolated network. E has JNR iff for any ǫ > 0 there is a sequence ( E ǫ n ) covering E such that for any n ∈ N and any x ∈ E ǫ n there is an w -open neighbourhood x ∈ U with diam ( U ∩ E ǫ n ) < ǫ . WCG ⇒ LUR ⇒ Kadec ⇒ JNR ⇔ descriptive. Concrete spaces C ( K ) with Kadec renorming: K - dyadic compacta, compact linearly ordered spaces, Valdivia compacta (hence Corson compacta), all ”cubes” [0 , 1] κ , AU-compacta.... JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
C ( K ) has JNR C -property (= C ( K ) has JNR -property + C p ( K ) is perfect) iff there exists a σ -discrete family in C p ( K ) which is a network in C ( K ) [Marciszewski-Pol]. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
C ( K ) has JNR C -property (= C ( K ) has JNR -property + C p ( K ) is perfect) iff there exists a σ -discrete family in C p ( K ) which is a network in C ( K ) [Marciszewski-Pol]. Concrete K : separable dyadic compacta, separable compact linearly ordered spaces.... [M.-P.]. Then C p ( K ) and C ( K ) w are σ -spaces (not ℵ -spaces). JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
There are (separable) compact K s.t. C p ( K ) are not σ -spaces. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
There are (separable) compact K s.t. C p ( K ) are not σ -spaces. If C p ( K ) is a σ -space ⇒ K is separable. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
There are (separable) compact K s.t. C p ( K ) are not σ -spaces. If C p ( K ) is a σ -space ⇒ K is separable. If E w is a σ -space ⇒ E is descriptive. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
There are (separable) compact K s.t. C p ( K ) are not σ -spaces. If C p ( K ) is a σ -space ⇒ K is separable. If E w is a σ -space ⇒ E is descriptive. E is descriptive � E w is a σ -space. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
There are (separable) compact K s.t. C p ( K ) are not σ -spaces. If C p ( K ) is a σ -space ⇒ K is separable. If E w is a σ -space ⇒ E is descriptive. E is descriptive � E w is a σ -space. Take E := C ( K ) with K := [0 , ω 1 ]. E is descriptive, so E w has a σ -isolated network, E w does not admit a σ -discrete network (since E w has uncountable pseudocharacter). Another example K separable: C ( K ( ω <ω )) over AU -compact K ( ω <ω ) := ω <ω ∪ ω ω ∪ {∞} [M.-P. 2009]: JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
There are (separable) compact K s.t. C p ( K ) are not σ -spaces. If C p ( K ) is a σ -space ⇒ K is separable. If E w is a σ -space ⇒ E is descriptive. E is descriptive � E w is a σ -space. Take E := C ( K ) with K := [0 , ω 1 ]. E is descriptive, so E w has a σ -isolated network, E w does not admit a σ -discrete network (since E w has uncountable pseudocharacter). Another example K separable: C ( K ( ω <ω )) over AU -compact K ( ω <ω ) := ω <ω ∪ ω ω ∪ {∞} [M.-P. 2009]: Kadec ⇒ JNR -property � JNR c -property. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
There are (separable) compact K s.t. C p ( K ) are not σ -spaces. If C p ( K ) is a σ -space ⇒ K is separable. If E w is a σ -space ⇒ E is descriptive. E is descriptive � E w is a σ -space. Take E := C ( K ) with K := [0 , ω 1 ]. E is descriptive, so E w has a σ -isolated network, E w does not admit a σ -discrete network (since E w has uncountable pseudocharacter). Another example K separable: C ( K ( ω <ω )) over AU -compact K ( ω <ω ) := ω <ω ∪ ω ω ∪ {∞} [M.-P. 2009]: Kadec ⇒ JNR -property � JNR c -property. C ( β N ) not descriptive. C p ( β N ), C p ( β N \ N ) are not σ -spaces. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
It is consistent with ZFC: there is a compact separable scattered space K such that C ( K ) has no Kadec renorming and C p ( K ) is not a σ -space. [M.-P.] JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
It is consistent with ZFC: there is a compact separable scattered space K such that C ( K ) has no Kadec renorming and C p ( K ) is not a σ -space. [M.-P.] Problem 20 (M-O-T-V) Does there exist E for which E w has a σ -isolated network and E has no Kadec renorming? JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
It is consistent with ZFC: there is a compact separable scattered space K such that C ( K ) has no Kadec renorming and C p ( K ) is not a σ -space. [M.-P.] Problem 20 (M-O-T-V) Does there exist E for which E w has a σ -isolated network and E has no Kadec renorming? Problem 21 Let E w be σ -space (or even an ℵ -space). Does E admit an equivalent Kadec norm? Describe those Banach spaces whose E w is a σ -space. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
It is consistent with ZFC: there is a compact separable scattered space K such that C ( K ) has no Kadec renorming and C p ( K ) is not a σ -space. [M.-P.] Problem 20 (M-O-T-V) Does there exist E for which E w has a σ -isolated network and E has no Kadec renorming? Problem 21 Let E w be σ -space (or even an ℵ -space). Does E admit an equivalent Kadec norm? Describe those Banach spaces whose E w is a σ -space. Problem 22 Describe (separable) compact K for which C ( K ) w is a σ -space. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Ascoli spaces. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Ascoli spaces. X is a k R -space if any real-valued map f on X is continuous, whenever f | K for any compact K ⊂ X is continuous. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Ascoli spaces. X is a k R -space if any real-valued map f on X is continuous, whenever f | K for any compact K ⊂ X is continuous. X is a s R -space if every real-valued sequentially continuous map on X is continuous. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Ascoli spaces. X is a k R -space if any real-valued map f on X is continuous, whenever f | K for any compact K ⊂ X is continuous. X is a s R -space if every real-valued sequentially continuous map on X is continuous. Theorem 23 (Pytkeev) C p ( X ) is Fr´ echet-Urysohn iff C p ( X ) is sequential iff C p ( X ) is a k-space. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Ascoli spaces. X is a k R -space if any real-valued map f on X is continuous, whenever f | K for any compact K ⊂ X is continuous. X is a s R -space if every real-valued sequentially continuous map on X is continuous. Theorem 23 (Pytkeev) C p ( X ) is Fr´ echet-Urysohn iff C p ( X ) is sequential iff C p ( X ) is a k-space. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Ascoli spaces. X is a k R -space if any real-valued map f on X is continuous, whenever f | K for any compact K ⊂ X is continuous. X is a s R -space if every real-valued sequentially continuous map on X is continuous. Theorem 23 (Pytkeev) C p ( X ) is Fr´ echet-Urysohn iff C p ( X ) is sequential iff C p ( X ) is a k-space. If C p ( X ) is angelic then C p ( X ) is a k R -space iff it is a s R -space. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
X is an Ascoli space if each compact K ⊂ C k ( X ) is evenly continuous [Banakh-Gabriyelyan], i.e. for ψ : X × C k ( X ) → R , ψ ( x , f ) := f ( x ), the map ψ | X × K is jointly continuous. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
X is an Ascoli space if each compact K ⊂ C k ( X ) is evenly continuous [Banakh-Gabriyelyan], i.e. for ψ : X × C k ( X ) → R , ψ ( x , f ) := f ( x ), the map ψ | X × K is jointly continuous. k -space ⇒ k R -space ⇒ Ascoli space. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
X is an Ascoli space if each compact K ⊂ C k ( X ) is evenly continuous [Banakh-Gabriyelyan], i.e. for ψ : X × C k ( X ) → R , ψ ( x , f ) := f ( x ), the map ψ | X × K is jointly continuous. k -space ⇒ k R -space ⇒ Ascoli space. Ascoli � k R -space. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
X is an Ascoli space if each compact K ⊂ C k ( X ) is evenly continuous [Banakh-Gabriyelyan], i.e. for ψ : X × C k ( X ) → R , ψ ( x , f ) := f ( x ), the map ψ | X × K is jointly continuous. k -space ⇒ k R -space ⇒ Ascoli space. Ascoli � k R -space. X is Ascoli iff the canonical evaluation map X ֒ → C k ( C k ( X )) is an embedding [Banakh-Gabriyelyan]. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
X is an Ascoli space if each compact K ⊂ C k ( X ) is evenly continuous [Banakh-Gabriyelyan], i.e. for ψ : X × C k ( X ) → R , ψ ( x , f ) := f ( x ), the map ψ | X × K is jointly continuous. k -space ⇒ k R -space ⇒ Ascoli space. Ascoli � k R -space. X is Ascoli iff the canonical evaluation map X ֒ → C k ( C k ( X )) is an embedding [Banakh-Gabriyelyan]. For an Ascoli space X the Ascoli’s theorem holds for C k ( X ). JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
X is an Ascoli space if each compact K ⊂ C k ( X ) is evenly continuous [Banakh-Gabriyelyan], i.e. for ψ : X × C k ( X ) → R , ψ ( x , f ) := f ( x ), the map ψ | X × K is jointly continuous. k -space ⇒ k R -space ⇒ Ascoli space. Ascoli � k R -space. X is Ascoli iff the canonical evaluation map X ֒ → C k ( C k ( X )) is an embedding [Banakh-Gabriyelyan]. For an Ascoli space X the Ascoli’s theorem holds for C k ( X ). Theorem 24 (Gabriyelyan-K.-Plebanek) E w is Ascoli iff E is finite-dimensional. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Problem 25 Does there exist a Banach space E containing a copy of ℓ 1 such that B w is Ascoli or even a k R -space? JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Problem 25 Does there exist a Banach space E containing a copy of ℓ 1 such that B w is Ascoli or even a k R -space? Theorem 26 (Gabriyelyan-K.-Plebanek) The following are equivalent for a Banach space E. (i) B w is Ascoli, i.e. B w embeds into C k ( C k ( B w )) ; (ii) B w is a k R -space; (iii) B w is a s R -space; (iv) E does not contain a copy of ℓ 1 . JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
What about Ascoli spaces C p ( X ) and C k ( X ) ? JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
What about Ascoli spaces C p ( X ) and C k ( X ) ? Theorem 27 (Gabriyelyan-Grebik-K.-Zdomskyy) Let X be a ˇ Cech-complete space. Then: (i) If C p ( X ) is Ascoli, then X is scattered. (ii) If X is scattered and stratifiable, then C p ( X ) is an Ascoli space. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
What about Ascoli spaces C p ( X ) and C k ( X ) ? Theorem 27 (Gabriyelyan-Grebik-K.-Zdomskyy) Let X be a ˇ Cech-complete space. Then: (i) If C p ( X ) is Ascoli, then X is scattered. (ii) If X is scattered and stratifiable, then C p ( X ) is an Ascoli space. Corollary 28 Let X be a completely metrizable space. Then C p ( X ) is Ascoli iff X is scattered. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Corollary 29 (A) For ˇ Cech-complete Lindel¨ of X, the following are equiv. (i) C p ( X ) is Ascoli. (ii) C p ( X ) is Fr´ echet–Urysohn. (iii) C p ( X ) is a k R -space. (iv) X is scattered. (B) If X is locally compact, then C p ( X ) is Ascoli iff X scattered. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Theorem 30 (Gabriyelyan-Grebik-K.-Zdomskyy) For paracompact of point-countable type X the following are equiv. (i) X is locally compact. (ii) C k ( X ) is a k R -space. (iii) C k ( X ) is an Ascoli space. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Theorem 30 (Gabriyelyan-Grebik-K.-Zdomskyy) For paracompact of point-countable type X the following are equiv. (i) X is locally compact. (ii) C k ( X ) is a k R -space. (iii) C k ( X ) is an Ascoli space. The space C p ([0 , ω 1 )) is Ascoli but not a k R -space. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Theorem 30 (Gabriyelyan-Grebik-K.-Zdomskyy) For paracompact of point-countable type X the following are equiv. (i) X is locally compact. (ii) C k ( X ) is a k R -space. (iii) C k ( X ) is an Ascoli space. The space C p ([0 , ω 1 )) is Ascoli but not a k R -space. (i) The first claim follows from the local compactness and the scattered property of [0 , ω 1 ). JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
Theorem 30 (Gabriyelyan-Grebik-K.-Zdomskyy) For paracompact of point-countable type X the following are equiv. (i) X is locally compact. (ii) C k ( X ) is a k R -space. (iii) C k ( X ) is an Ascoli space. The space C p ([0 , ω 1 )) is Ascoli but not a k R -space. (i) The first claim follows from the local compactness and the scattered property of [0 , ω 1 ). (ii) Assume E := C p ([0 , ω 1 )) is a k R -space. Since [0 , ω 1 ) is pseudocompat, E is dominated by a Banach topology. Hence E is angelic, so every compact set in E is Fr´ echet-Urysohn. Therefore E is a s R -space, and then [0 , ω 1 ) is realcompact, a contradiction. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
When E w is stratifiable ? JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
When E w is stratifiable ? X is stratifiable iff to each open U ⊂ X one can assign a continuous function f U : X → [0 , 1] such that f − 1 (0) = X \ U , and f U ≤ f V whenever U ⊂ V [Borges]. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces
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