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Two Selected Topics on the weak topology of Banach spaces

JERZY KA ¸KOL

• A. MICKIEWICZ UNIVERSITY, POZNA´

N, AND CZECH ACADEMY OF SCIENCES, PRAHA

B¸ edlewo 2016

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

E - Banach space, Ew:=(E, w(E, E ′)), Bw – the closed unit ball with the weak topology, K – compact space.

1 Cosmic spaces, ℵ0-spaces, ℵ-spaces and σ-spaces,

topological characterizations.

2 Networks for spaces Ew; general case. 3 Networks for spaces Ew where E := C(K). 4 Generalized metric concepts for spaces Ew and Bw;

a bit of history.

5

kR-spaces, Ascoli and stratifiable spaces Ew and Bw.

6 Ascoli spaces Cp(X) and Ck(X). 7 Open problems. JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

Some Motivation.

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

Some Motivation. ”Surprisingly enough tools coming from pure set-theoretical topology, like the concept of network, are of great importance to study successfully renorming theory in Banach spaces” [Cascales-Orihuela], Recent Progress in Topology III (2013). Chapter of book.

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

Some Motivation. ”Surprisingly enough tools coming from pure set-theoretical topology, like the concept of network, are of great importance to study successfully renorming theory in Banach spaces” [Cascales-Orihuela], Recent Progress in Topology III (2013). Chapter of book. E admits an equivalent LUR norm iff Ew has a σ-slicely isolated network. [Molto-Orihuela-Troyanski-Valdivia] in A nonlinear Transfer Technique for Renorming, Springer (2009).

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

Some Motivation. ”Surprisingly enough tools coming from pure set-theoretical topology, like the concept of network, are of great importance to study successfully renorming theory in Banach spaces” [Cascales-Orihuela], Recent Progress in Topology III (2013). Chapter of book. E admits an equivalent LUR norm iff Ew has a σ-slicely isolated network. [Molto-Orihuela-Troyanski-Valdivia] in A nonlinear Transfer Technique for Renorming, Springer (2009). An excellent monograph of renorming theory up to 1993 is: [Deville, Godefroy, Zizler] Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics.

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

For a Banach space E the following are equivalent: (i) Every Ew-compact set is Ew-metrizable. (ii) Every Ew-compact set is contained in a separable subset of Ew. (iii) Ew is the image under a compact-covering map of a metric space F.

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

For a Banach space E the following are equivalent: (i) Every Ew-compact set is Ew-metrizable. (ii) Every Ew-compact set is contained in a separable subset of Ew. (iii) Ew is the image under a compact-covering map of a metric space F. If, for example, E ′ is w ∗-separable (equiv., there is a continuous injection E ֒ → ℓ∞), (iii) holds but F need not be

• separable. Take E := C[0, 1].

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

For a Banach space E the following are equivalent: (i) Every Ew-compact set is Ew-metrizable. (ii) Every Ew-compact set is contained in a separable subset of Ew. (iii) Ew is the image under a compact-covering map of a metric space F. If, for example, E ′ is w ∗-separable (equiv., there is a continuous injection E ֒ → ℓ∞), (iii) holds but F need not be

• separable. Take E := C[0, 1].

Theorem 1 (Michael) For a regular space X the following are equivalent.

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

For a Banach space E the following are equivalent: (i) Every Ew-compact set is Ew-metrizable. (ii) Every Ew-compact set is contained in a separable subset of Ew. (iii) Ew is the image under a compact-covering map of a metric space F. If, for example, E ′ is w ∗-separable (equiv., there is a continuous injection E ֒ → ℓ∞), (iii) holds but F need not be

• separable. Take E := C[0, 1].

Theorem 1 (Michael) For a regular space X the following are equivalent. (i) X is the image under a compact-covering map of a separable metric space.

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

For a Banach space E the following are equivalent: (i) Every Ew-compact set is Ew-metrizable. (ii) Every Ew-compact set is contained in a separable subset of Ew. (iii) Ew is the image under a compact-covering map of a metric space F. If, for example, E ′ is w ∗-separable (equiv., there is a continuous injection E ֒ → ℓ∞), (iii) holds but F need not be

• separable. Take E := C[0, 1].

Theorem 1 (Michael) For a regular space X the following are equivalent. (i) X is the image under a compact-covering map of a separable metric space. (ii) There exists a countable family D (countable k-network)

• f subsets in X such that for each open set U in X and

compact K ⊂ U there exists D ∈ D with K ⊂ D ⊂ U.

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

Few definitions and facts. X – regular.

1 X is an ℵ0-space if X has a countable k-network

[Michael]. Any metric separable X is an ℵ0-space.

2 X is cosmic if X has a countable network. 3 X is cosmic iff X is a continuous image of a metric

separable space.

4 X is an ℵ-space if X has a σ-locally finite k-network

[0’Meara]. Any metric space is an ℵ-space, compact sets in ℵ-spaces are metrizable, see Gruenhage’s works.

5 X is an ℵ0-space iff X is a Lindel¨

• f ℵ-space.

6 X is a σ-space if X has a σ-locally finite network

[Okuyama] (eq., σ-discrete network [Siwiec-Nagata]).

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 2 (0’Meara-Foged) If X is an ℵ0-space and Y is a (paracompact) ℵ-space, then Ck(X, Y ) with the compact-open topology is an (paracompact) ℵ-space. Hence, if X is separable metric and Y is metric, then Ck(X, Y ) is paracompact.

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

Theorem 2 (0’Meara-Foged) If X is an ℵ0-space and Y is a (paracompact) ℵ-space, then Ck(X, Y ) with the compact-open topology is an (paracompact) ℵ-space. Hence, if X is separable metric and Y is metric, then Ck(X, Y ) is paracompact. Theorem 3 (Michael, Sakai) Cp(X) is an ℵ0-space iff X is countable iff Cp(X) is an ℵ-space.

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

Theorem 2 (0’Meara-Foged) If X is an ℵ0-space and Y is a (paracompact) ℵ-space, then Ck(X, Y ) with the compact-open topology is an (paracompact) ℵ-space. Hence, if X is separable metric and Y is metric, then Ck(X, Y ) is paracompact. Theorem 3 (Michael, Sakai) Cp(X) is an ℵ0-space iff X is countable iff Cp(X) is an ℵ-space. Theorem 4 (Gabriyelyan-K.-Kubi´ s-Marciszewski) An ℵ-space X is metrizable iff X is Fr´ echet-Urysohn with α4-property. Hence a Fr´ echet-Urysohn topological group is metrizable iff it is an ℵ-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

separable metrizable

• ℵ0-space
• cosmic
• metrizable

strict ℵ-space

• strict σ-space
• ℵ-space

σ-space.

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

How to describe the topology of cosmic....ℵ0-spaces?

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

How to describe the topology of cosmic....ℵ0-spaces? Endow NN with the order, i.e., α ≤ β if αi ≤ βi for all i ∈ N, α = (αi)i∈N, β = (βi)i∈N. For every α ∈ NN, k ∈ N, set Ik(α) :=

• β ∈ NN : βi = αi for i = 1, . . . , k
• .

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

How to describe the topology of cosmic....ℵ0-spaces? Endow NN with the order, i.e., α ≤ β if αi ≤ βi for all i ∈ N, α = (αi)i∈N, β = (βi)i∈N. For every α ∈ NN, k ∈ N, set Ik(α) :=

• β ∈ NN : βi = αi for i = 1, . . . , k
• .

Let M ⊆ NN and U = {Uα : α ∈ M} be an M-decreasing family of subsets of a set X. Define the countable family DU

• f subsets of X by

DU := {Dk(α) : α ∈ M, k ∈ N}, where Dk(α) :=

• β∈Ik(α)∩M

Uβ, U satisfies condition (D) if Uα =

k∈N Dk(α), α ∈ M.

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

How to describe the topology of cosmic....ℵ0-spaces? Endow NN with the order, i.e., α ≤ β if αi ≤ βi for all i ∈ N, α = (αi)i∈N, β = (βi)i∈N. For every α ∈ NN, k ∈ N, set Ik(α) :=

• β ∈ NN : βi = αi for i = 1, . . . , k
• .

Let M ⊆ NN and U = {Uα : α ∈ M} be an M-decreasing family of subsets of a set X. Define the countable family DU

• f subsets of X by

DU := {Dk(α) : α ∈ M, k ∈ N}, where Dk(α) :=

• β∈Ik(α)∩M

Uβ, U satisfies condition (D) if Uα =

k∈N Dk(α), α ∈ M.

(X, τ) has a small base if there exists an M-decreasing base

• f τ for some M ⊆ NN [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 = {Uα : α ∈ M} with condition (D). In that case the family DU is a countable network in X. (ii) X is an ℵ0-space iff X has a small base U = {Uα : α ∈ M} with condition (D) such that the family DU 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 = {Uα : α ∈ M} with condition (D). In that case the family DU is a countable network in X. (ii) X is an ℵ0-space iff X has a small base U = {Uα : α ∈ M} with condition (D) such that the family DU 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 Bw is an ℵ- and k-space?

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

When Bw 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) Bw is Fr´ echet–Urysohn. (ii) Bw is sequential. (iii) Bw is a k-space, i.e. P ⊂ Bw is closed in Bw if P ∩ K is closed in K for all compact K ⊂ Bw. (iv) E contains no isomorphic copy of ℓ1.

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

When Bw 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) Bw is Fr´ echet–Urysohn. (ii) Bw is sequential. (iii) Bw is a k-space, i.e. P ⊂ Bw is closed in Bw if P ∩ K is closed in K for all compact K ⊂ Bw. (iv) E contains no isomorphic copy of ℓ1. Theorem 8 (Schl¨ uchtermann-Wheeler) If E is a Banach space, then Ew 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) Bw is (separable) metrizable. (ii) Bw is an ℵ0-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) Bw is (separable) metrizable. (ii) Bw is an ℵ0-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) Bw is (separable) metrizable. (ii) Bw 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 Ew 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 Ew is an ℵ-space. Problem 11 Describe those Banach spaces E for which Ew 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 Ew is an ℵ-space. Problem 11 Describe those Banach spaces E for which Ew is an ℵ-space. Theorem 12 (Corson) C[0, 1]w is not an ℵ0-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 Ew is an ℵ-space. Problem 11 Describe those Banach spaces E for which Ew is an ℵ-space. Theorem 12 (Corson) C[0, 1]w is not an ℵ0-space. Theorem 13 (Gabriyelyan-K.-Kubi´ s-Marciszewski) For a Banach space E := C(K) the space Ew is an ℵ-space iff Ew is an ℵ0-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 ℓ1. The following conditions are equivalent: (i) Ew is an ℵ-space (ii) Ew is an ℵ0-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 ℓ1. The following conditions are equivalent: (i) Ew is an ℵ-space (ii) Ew is an ℵ0-space. (iii) The dual E ′ is separable. (ℓ1)w is an ℵ0-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 ℓ1. The following conditions are equivalent: (i) Ew is an ℵ-space (ii) Ew is an ℵ0-space. (iii) The dual E ′ is separable. (ℓ1)w is an ℵ0-space. (JT)w is a σ-space but not an ℵ-space. Corollary 15 If E is separable and does not contain ℓ1, then Ew is an ℵ0-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) (ℓ1(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) (ℓ1(Γ))w is an ℵ-space iff the cardinality of Γ does not exceed the continuum. ℓ1(Γ) with the weak topology does not have countable pseudocharacter whenever |Γ| > 2ℵ0.

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

Theorem 16 (Gabriyelyan-K.-Kubi´ s-Marciszewski) (ℓ1(Γ))w is an ℵ-space iff the cardinality of Γ does not exceed the continuum. ℓ1(Γ) with the weak topology does not have countable pseudocharacter whenever |Γ| > 2ℵ0. Hence (ℓ1(R))w is an ℵ-space which is not an ℵ0-space and (ℓ1(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) (ℓ1(Γ))w is an ℵ-space iff the cardinality of Γ does not exceed the continuum. ℓ1(Γ) with the weak topology does not have countable pseudocharacter whenever |Γ| > 2ℵ0. Hence (ℓ1(R))w is an ℵ-space which is not an ℵ0-space and (ℓ1(R))w is not normal. Last claim follows from: Theorem 17 (Reznichenko) Let E be a Banach space. Then Ew is Lindel¨

• f iff Ew is normal

iff Ew 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 Cp(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 Cp(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 Cp(X) and C(K)w are already characterized. Any σ-space is perfect [Gruenhage], so σ-spaces have countable pseudocharacter. If Ew 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 Cp(X) and C(K)w are already characterized. Any σ-space is perfect [Gruenhage], so σ-spaces have countable pseudocharacter. If Ew 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 := ℓp(Γ) with 1 < p < ∞. Then ψ(Ew) ≥ |Γ|, where Ew := (E, σ(E, E ′)). Hence ℓp(Γ)w are not σ-spaces for any uncountable Γ. More: Ew for any nonseparable weakly Lindel¨

• f 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 Cp(X) and C(K)w are already characterized. Any σ-space is perfect [Gruenhage], so σ-spaces have countable pseudocharacter. If Ew 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 := ℓp(Γ) with 1 < p < ∞. Then ψ(Ew) ≥ |Γ|, where Ew := (E, σ(E, E ′)). Hence ℓp(Γ)w are not σ-spaces for any uncountable Γ. More: Ew for any nonseparable weakly Lindel¨

• f 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 Ew) iff E has the JNR-property iff Ew 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 Ew) iff E has the JNR-property iff Ew 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 Ew) iff E has the JNR-property iff Ew 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 JNRC-property (= C(K) has JNR-property + Cp(K) is perfect) iff there exists a σ-discrete family in Cp(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 JNRC-property (= C(K) has JNR-property + Cp(K) is perfect) iff there exists a σ-discrete family in Cp(K) which is a network in C(K) [Marciszewski-Pol]. Concrete K: separable dyadic compacta, separable compact linearly ordered spaces.... [M.-P.]. Then Cp(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. Cp(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. Cp(K) are not σ-spaces. If Cp(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. Cp(K) are not σ-spaces. If Cp(K) is a σ-space ⇒ K is separable. If Ew 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. Cp(K) are not σ-spaces. If Cp(K) is a σ-space ⇒ K is separable. If Ew is a σ-space ⇒ E is descriptive. E is descriptive Ew is a σ-space.

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

There are (separable) compact K s.t. Cp(K) are not σ-spaces. If Cp(K) is a σ-space ⇒ K is separable. If Ew is a σ-space ⇒ E is descriptive. E is descriptive Ew is a σ-space. Take E := C(K) with K := [0, ω1]. E is descriptive, so Ew has a σ-isolated network, Ew does not admit a σ-discrete network (since Ew has uncountable pseudocharacter). Another example K separable: C(K(ω<ω)) over AU-compact K(ω<ω) := ω<ω ∪ ωω ∪ {∞} [M.-P. 2009]:

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

There are (separable) compact K s.t. Cp(K) are not σ-spaces. If Cp(K) is a σ-space ⇒ K is separable. If Ew is a σ-space ⇒ E is descriptive. E is descriptive Ew is a σ-space. Take E := C(K) with K := [0, ω1]. E is descriptive, so Ew has a σ-isolated network, Ew does not admit a σ-discrete network (since Ew has uncountable pseudocharacter). Another example K separable: C(K(ω<ω)) over AU-compact K(ω<ω) := ω<ω ∪ ωω ∪ {∞} [M.-P. 2009]: Kadec ⇒ JNR-property JNRc-property.

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

There are (separable) compact K s.t. Cp(K) are not σ-spaces. If Cp(K) is a σ-space ⇒ K is separable. If Ew is a σ-space ⇒ E is descriptive. E is descriptive Ew is a σ-space. Take E := C(K) with K := [0, ω1]. E is descriptive, so Ew has a σ-isolated network, Ew does not admit a σ-discrete network (since Ew has uncountable pseudocharacter). Another example K separable: C(K(ω<ω)) over AU-compact K(ω<ω) := ω<ω ∪ ωω ∪ {∞} [M.-P. 2009]: Kadec ⇒ JNR-property JNRc-property. C(βN) not descriptive. Cp(βN), Cp(βN \ 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 Cp(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 Cp(K) is not a σ-space. [M.-P.] Problem 20 (M-O-T-V) Does there exist E for which Ew 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 Cp(K) is not a σ-space. [M.-P.] Problem 20 (M-O-T-V) Does there exist E for which Ew has a σ-isolated network and E has no Kadec renorming? Problem 21 Let Ew be σ-space (or even an ℵ-space). Does E admit an equivalent Kadec norm? Describe those Banach spaces whose Ew 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 Cp(K) is not a σ-space. [M.-P.] Problem 20 (M-O-T-V) Does there exist E for which Ew has a σ-isolated network and E has no Kadec renorming? Problem 21 Let Ew be σ-space (or even an ℵ-space). Does E admit an equivalent Kadec norm? Describe those Banach spaces whose Ew 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 kR-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 kR-space if any real-valued map f on X is continuous, whenever f |K for any compact K ⊂ X is continuous. X is a sR-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 kR-space if any real-valued map f on X is continuous, whenever f |K for any compact K ⊂ X is continuous. X is a sR-space if every real-valued sequentially continuous map on X is continuous. Theorem 23 (Pytkeev) Cp(X) is Fr´ echet-Urysohn iff Cp(X) is sequential iff Cp(X) is a k-space.

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

Ascoli spaces. X is a kR-space if any real-valued map f on X is continuous, whenever f |K for any compact K ⊂ X is continuous. X is a sR-space if every real-valued sequentially continuous map on X is continuous. Theorem 23 (Pytkeev) Cp(X) is Fr´ echet-Urysohn iff Cp(X) is sequential iff Cp(X) is a k-space.

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

Ascoli spaces. X is a kR-space if any real-valued map f on X is continuous, whenever f |K for any compact K ⊂ X is continuous. X is a sR-space if every real-valued sequentially continuous map on X is continuous. Theorem 23 (Pytkeev) Cp(X) is Fr´ echet-Urysohn iff Cp(X) is sequential iff Cp(X) is a k-space. If Cp(X) is angelic then Cp(X) is a kR-space iff it is a sR-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 ⊂ Ck(X) is evenly continuous [Banakh-Gabriyelyan], i.e. for ψ : X × Ck(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 ⊂ Ck(X) is evenly continuous [Banakh-Gabriyelyan], i.e. for ψ : X × Ck(X) → R, ψ(x, f ) := f (x), the map ψ|X × K is jointly continuous. k-space ⇒ kR-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 ⊂ Ck(X) is evenly continuous [Banakh-Gabriyelyan], i.e. for ψ : X × Ck(X) → R, ψ(x, f ) := f (x), the map ψ|X × K is jointly continuous. k-space ⇒ kR-space ⇒ Ascoli space. Ascoli kR-space.

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

X is an Ascoli space if each compact K ⊂ Ck(X) is evenly continuous [Banakh-Gabriyelyan], i.e. for ψ : X × Ck(X) → R, ψ(x, f ) := f (x), the map ψ|X × K is jointly continuous. k-space ⇒ kR-space ⇒ Ascoli space. Ascoli kR-space. X is Ascoli iff the canonical evaluation map X ֒ → Ck(Ck(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 ⊂ Ck(X) is evenly continuous [Banakh-Gabriyelyan], i.e. for ψ : X × Ck(X) → R, ψ(x, f ) := f (x), the map ψ|X × K is jointly continuous. k-space ⇒ kR-space ⇒ Ascoli space. Ascoli kR-space. X is Ascoli iff the canonical evaluation map X ֒ → Ck(Ck(X)) is an embedding [Banakh-Gabriyelyan]. For an Ascoli space X the Ascoli’s theorem holds for Ck(X).

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

X is an Ascoli space if each compact K ⊂ Ck(X) is evenly continuous [Banakh-Gabriyelyan], i.e. for ψ : X × Ck(X) → R, ψ(x, f ) := f (x), the map ψ|X × K is jointly continuous. k-space ⇒ kR-space ⇒ Ascoli space. Ascoli kR-space. X is Ascoli iff the canonical evaluation map X ֒ → Ck(Ck(X)) is an embedding [Banakh-Gabriyelyan]. For an Ascoli space X the Ascoli’s theorem holds for Ck(X). Theorem 24 (Gabriyelyan-K.-Plebanek) Ew 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 ℓ1 such that Bw is Ascoli or even a kR-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 ℓ1 such that Bw is Ascoli or even a kR-space? Theorem 26 (Gabriyelyan-K.-Plebanek) The following are equivalent for a Banach space E. (i) Bw is Ascoli, i.e. Bw embeds into Ck(Ck(Bw)); (ii) Bw is a kR-space; (iii) Bw is a sR-space; (iv) E does not contain a copy of ℓ1.

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

What about Ascoli spaces Cp(X) and Ck(X) ?

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

What about Ascoli spaces Cp(X) and Ck(X) ? Theorem 27 (Gabriyelyan-Grebik-K.-Zdomskyy) Let X be a ˇ Cech-complete space. Then: (i) If Cp(X) is Ascoli, then X is scattered. (ii) If X is scattered and stratifiable, then Cp(X) is an Ascoli space.

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

What about Ascoli spaces Cp(X) and Ck(X) ? Theorem 27 (Gabriyelyan-Grebik-K.-Zdomskyy) Let X be a ˇ Cech-complete space. Then: (i) If Cp(X) is Ascoli, then X is scattered. (ii) If X is scattered and stratifiable, then Cp(X) is an Ascoli space. Corollary 28 Let X be a completely metrizable space. Then Cp(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¨

• f X, the following are equiv.

(i) Cp(X) is Ascoli. (ii) Cp(X) is Fr´ echet–Urysohn. (iii) Cp(X) is a kR-space. (iv) X is scattered. (B) If X is locally compact, then Cp(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) Ck(X) is a kR-space. (iii) Ck(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) Ck(X) is a kR-space. (iii) Ck(X) is an Ascoli space. The space Cp([0, ω1)) is Ascoli but not a kR-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) Ck(X) is a kR-space. (iii) Ck(X) is an Ascoli space. The space Cp([0, ω1)) is Ascoli but not a kR-space. (i) The first claim follows from the local compactness and the scattered property of [0, ω1).

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) Ck(X) is a kR-space. (iii) Ck(X) is an Ascoli space. The space Cp([0, ω1)) is Ascoli but not a kR-space. (i) The first claim follows from the local compactness and the scattered property of [0, ω1). (ii) Assume E := Cp([0, ω1)) is a kR-space. Since [0, ω1) 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 sR-space, and then [0, ω1) is realcompact, a contradiction.

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

When Ew is stratifiable?

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

When Ew is stratifiable? X is stratifiable iff to each open U ⊂ X one can assign a continuous function fU : X → [0, 1] such that f −1(0) = X \ U, and fU ≤ fV whenever U ⊂ V [Borges].

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

When Ew is stratifiable? X is stratifiable iff to each open U ⊂ X one can assign a continuous function fU : X → [0, 1] such that f −1(0) = X \ U, and fU ≤ fV whenever U ⊂ V [Borges]. Metrizable ⇒ stratifiable ⇒ paracompact+ closed sets are Gδ.

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

When Ew is stratifiable? X is stratifiable iff to each open U ⊂ X one can assign a continuous function fU : X → [0, 1] such that f −1(0) = X \ U, and fU ≤ fV whenever U ⊂ V [Borges]. Metrizable ⇒ stratifiable ⇒ paracompact+ closed sets are Gδ. Fr´ echet-Urysohn ℵ-space ⇒ stratifiable [Foged] ⇒ σ-space [Gruenhage].

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

When Ew is stratifiable? X is stratifiable iff to each open U ⊂ X one can assign a continuous function fU : X → [0, 1] such that f −1(0) = X \ U, and fU ≤ fV whenever U ⊂ V [Borges]. Metrizable ⇒ stratifiable ⇒ paracompact+ closed sets are Gδ. Fr´ echet-Urysohn ℵ-space ⇒ stratifiable [Foged] ⇒ σ-space [Gruenhage]. If X is stratifiable, then X is separable iff X is Lindel¨

• f iff X

has countable network.

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

When Ew is stratifiable? X is stratifiable iff to each open U ⊂ X one can assign a continuous function fU : X → [0, 1] such that f −1(0) = X \ U, and fU ≤ fV whenever U ⊂ V [Borges]. Metrizable ⇒ stratifiable ⇒ paracompact+ closed sets are Gδ. Fr´ echet-Urysohn ℵ-space ⇒ stratifiable [Foged] ⇒ σ-space [Gruenhage]. If X is stratifiable, then X is separable iff X is Lindel¨

• f iff X

has countable network. X stratifiable, A ⊂ X closed, then there is a continuous linear extender e : Ck(A) → Ck(X), e(f )|A = f for any f ∈ C(A), (Dugundji extenstion property) [Borges].

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 Polish ⇒ Ck(X) is stratifiable [Gartside-Reznichenko]. They conjectured: If X is separable metrizable and Ck(X) is stratifiable, then X is Polish.

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

X is Polish ⇒ Ck(X) is stratifiable [Gartside-Reznichenko]. They conjectured: If X is separable metrizable and Ck(X) is stratifiable, then X is Polish. X separable metrizable and Ck(X) stratifiable ⇒ X contains a dense Polish subspace [Reznichenko-Tsaban-Zdomskyy].

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

X is Polish ⇒ Ck(X) is stratifiable [Gartside-Reznichenko]. They conjectured: If X is separable metrizable and Ck(X) is stratifiable, then X is Polish. X separable metrizable and Ck(X) stratifiable ⇒ X contains a dense Polish subspace [Reznichenko-Tsaban-Zdomskyy]. If X is metrizable and separable, then Ck(X) is stratifiable iff X is Polish [Reznichenko].

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

X is Polish ⇒ Ck(X) is stratifiable [Gartside-Reznichenko]. They conjectured: If X is separable metrizable and Ck(X) is stratifiable, then X is Polish. X separable metrizable and Ck(X) stratifiable ⇒ X contains a dense Polish subspace [Reznichenko-Tsaban-Zdomskyy]. If X is metrizable and separable, then Ck(X) is stratifiable iff X is Polish [Reznichenko]. Cp(X) is stratifiable iff X is countable [Gartside].

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

X is Polish ⇒ Ck(X) is stratifiable [Gartside-Reznichenko]. They conjectured: If X is separable metrizable and Ck(X) is stratifiable, then X is Polish. X separable metrizable and Ck(X) stratifiable ⇒ X contains a dense Polish subspace [Reznichenko-Tsaban-Zdomskyy]. If X is metrizable and separable, then Ck(X) is stratifiable iff X is Polish [Reznichenko]. Cp(X) is stratifiable iff X is countable [Gartside]. Many examples of nonmetrizable stratifiable LCS are provided by [Shkarin].

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 31 (Gartside) Ew is stratifiable iff E is finite-dimensional.

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

Theorem 31 (Gartside) Ew is stratifiable iff E is finite-dimensional. Theorem 32 (Corson-Lindenstrauss) For Bw of a nonseparable Hilbert space E and any 0 < α < β there exists no weak-continuous retraction r : βBw → αBw, i.e. a map r such that r(x) = x for every x ∈ αBw. Theorem 33 (Aviles-Marciszewski) For a nonseparable Hilbert space E and any 0 < α < β there is no continuous extender T : C(αBw) → C(βBw).

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

Easier approach for a weaker result: If E is weakly Lindel¨

• f nonseparable, then Bw is not a σ-space (since Ew is

not a σ-space). Hence Bw is not stratifiable.

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

Easier approach for a weaker result: If E is weakly Lindel¨

• f nonseparable, then Bw is not a σ-space (since Ew is

not a σ-space). Hence Bw is not stratifiable. Problem 34 Characterize those Banach spaces E for which Bw is stratifiable (has the Dugundji extension property).

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces

Easier approach for a weaker result: If E is weakly Lindel¨

• f nonseparable, then Bw is not a σ-space (since Ew is

not a σ-space). Hence Bw is not stratifiable. Problem 34 Characterize those Banach spaces E for which Bw is stratifiable (has the Dugundji extension property). Problem 35 Is the ball Bw a stratifiable space for E := JT ?

JERZY KA ¸KOL Two Selected Topics on the weak topology of Banach spaces