On Banach spaces which are weak sequentially dense in its bidual - - PowerPoint PPT Presentation

On Banach spaces which are weak∗ sequentially dense in its bidual

Jos´ e Rodr´ ıguez

Universidad de Murcia

Workshop on Banach spaces and Banach lattices Madrid, September 9, 2019

Research supported by Agencia Estatal de Investigaci´

• n/FEDER (MTM2017-86182-P) and Fundaci´
• n S´

eneca (20797/PI/18)

Throughout this talk X is a Banach space.

Throughout this talk X is a Banach space. Goldstine’s theorem BX is w∗-dense in BX ∗∗.

Throughout this talk X is a Banach space. Goldstine’s theorem BX is w∗-dense in BX ∗∗. Therefore, X is w∗-dense in X ∗∗.

Throughout this talk X is a Banach space. Goldstine’s theorem BX is w∗-dense in BX ∗∗. Therefore, X is w∗-dense in X ∗∗. Notation X ∈ SD iff X is w∗-sequentially dense in X ∗∗.

Throughout this talk X is a Banach space. Goldstine’s theorem BX is w∗-dense in BX ∗∗. Therefore, X is w∗-dense in X ∗∗. Notation X ∈ SD iff X is w∗-sequentially dense in X ∗∗. X ∈ SD if X is reflexive (obvious)

Throughout this talk X is a Banach space. Goldstine’s theorem BX is w∗-dense in BX ∗∗. Therefore, X is w∗-dense in X ∗∗. Notation X ∈ SD iff X is w∗-sequentially dense in X ∗∗. X ∈ SD if X is reflexive (obvious) X ∗ is separable [ ⇐ ⇒ (BX ∗∗,w∗) is metrizable ]

Throughout this talk X is a Banach space. Goldstine’s theorem BX is w∗-dense in BX ∗∗. Therefore, X is w∗-dense in X ∗∗. Notation X ∈ SD iff X is w∗-sequentially dense in X ∗∗. X ∈ SD if X is reflexive (obvious) X ∗ is separable [ ⇐ ⇒ (BX ∗∗,w∗) is metrizable ] (BX ∗∗,w∗) is Fr´ echet-Urysohn = ⇒ X ∈ SD

Throughout this talk X is a Banach space. Goldstine’s theorem BX is w∗-dense in BX ∗∗. Therefore, X is w∗-dense in X ∗∗. Notation X ∈ SD iff X is w∗-sequentially dense in X ∗∗. X ∈ SD if X is reflexive (obvious) X ∗ is separable [ ⇐ ⇒ (BX ∗∗,w∗) is metrizable ] X ∈ SD if X is weakly sequentially complete and non-reflexive (e.g. ℓ1 and L1) (BX ∗∗,w∗) is Fr´ echet-Urysohn = ⇒ X ∈ SD

Throughout this talk X is a Banach space. Goldstine’s theorem BX is w∗-dense in BX ∗∗. Therefore, X is w∗-dense in X ∗∗. Notation X ∈ SD iff X is w∗-sequentially dense in X ∗∗. X ∈ SD if X is reflexive (obvious) X ∗ is separable [ ⇐ ⇒ (BX ∗∗,w∗) is metrizable ] X ∈ SD if X is weakly sequentially complete and non-reflexive (e.g. ℓ1 and L1) X = c0(Γ) for uncountable Γ (BX ∗∗,w∗) is Fr´ echet-Urysohn = ⇒ X ∈ SD

Throughout this talk X is a Banach space. Goldstine’s theorem BX is w∗-dense in BX ∗∗. Therefore, X is w∗-dense in X ∗∗. Notation X ∈ SD iff X is w∗-sequentially dense in X ∗∗. X ∈ SD if X is reflexive (obvious) X ∗ is separable [ ⇐ ⇒ (BX ∗∗,w∗) is metrizable ] X ∈ SD if X is weakly sequentially complete and non-reflexive (e.g. ℓ1 and L1) X = c0(Γ) for uncountable Γ (BX ∗∗,w∗) is Fr´ echet-Urysohn = ⇒ X ∈ SD = ⇒ X ⊃ ℓ1.

Throughout this talk X is a Banach space. Goldstine’s theorem BX is w∗-dense in BX ∗∗. Therefore, X is w∗-dense in X ∗∗. Notation X ∈ SD iff X is w∗-sequentially dense in X ∗∗. X ∈ SD if X is reflexive (obvious) X ∗ is separable [ ⇐ ⇒ (BX ∗∗,w∗) is metrizable ] X ∈ SD if X is weakly sequentially complete and non-reflexive (e.g. ℓ1 and L1) X = c0(Γ) for uncountable Γ (BX ∗∗,w∗) is Fr´ echet-Urysohn = ⇒ X ∈ SD = ⇒ X ⊃ ℓ1. Theorem (Odell-Rosenthal, Bourgain-Fremlin-Talagrand) Suppose X is separable. Then: (BX ∗∗,w∗) is Fr´ echet-Urysohn ⇐ ⇒ X ∈ SD ⇐ ⇒ X ⊃ ℓ1.

(BX ∗∗,w∗) is Fr´ echet-Urysohn X ∈ SD X ⊃ ℓ1

(BX ∗∗,w∗) is Fr´ echet-Urysohn X ∈ SD X ⊃ ℓ1 (BX ∗∗,w∗) is Corson

(BX ∗∗,w∗) is Fr´ echet-Urysohn X ∈ SD X ⊃ ℓ1 (BX ∗∗,w∗) is Corson X ∗ is WCG

(BX ∗∗,w∗) is Fr´ echet-Urysohn X ∈ SD X ⊃ ℓ1 (BX ∗∗,w∗) is Corson X ∗ is WCG X is Asplund

(BX ∗∗,w∗) is Fr´ echet-Urysohn X ∈ SD X ⊃ ℓ1 (BX ∗∗,w∗) is Corson X ∗ is WCG X is Asplund Theorem (Deville-Godefroy, Orihuela) (BX ∗∗,w∗) is Corson ⇐ ⇒ X ∈ SD and X is Asplund.

(BX ∗∗,w∗) is Fr´ echet-Urysohn X ∈ SD X ⊃ ℓ1 (BX ∗∗,w∗) is Corson X ∗ is WCG X is Asplund Theorem (Deville-Godefroy, Orihuela) (BX ∗∗,w∗) is Corson ⇐ ⇒ X ∈ SD and X is Asplund. A Banach lattice X is Asplund if and only if X ⊃ ℓ1.

(BX ∗∗,w∗) is Fr´ echet-Urysohn X ∈ SD X ⊃ ℓ1 (BX ∗∗,w∗) is Corson X ∗ is WCG X is Asplund Theorem (Deville-Godefroy, Orihuela) (BX ∗∗,w∗) is Corson ⇐ ⇒ X ∈ SD and X is Asplund. A Banach lattice X is Asplund if and only if X ⊃ ℓ1. Corollary Suppose X is a Banach lattice. Then (BX ∗∗,w∗) is Corson ⇐ ⇒ X ∈ SD.

(BX ∗∗,w∗) is Fr´ echet-Urysohn X ∈ SD X ⊃ ℓ1 (BX ∗∗,w∗) is Corson X ∗ is WCG X is Asplund Theorem (Deville-Godefroy, Orihuela) (BX ∗∗,w∗) is Corson ⇐ ⇒ X ∈ SD and X is Asplund. A Banach lattice X is Asplund if and only if X ⊃ ℓ1. Corollary Suppose X is a Banach lattice. Then (BX ∗∗,w∗) is Corson ⇐ ⇒ X ∈ SD. Question X ∈ SD = ⇒ (BX ∗∗,w∗) is Fr´ echet-Urysohn ???

“Vague” question Which topological properties does (BX ∗,w∗) enjoy whenever X ∈ SD ???

“Vague” question Which topological properties does (BX ∗,w∗) enjoy whenever X ∈ SD ??? Given a set A ⊂ X ∗, we write S1(A) :=

• x∗ ∈ X ∗ : ∃(x∗

n) ⊂ A such that w∗- lim n→∞x∗ n = x∗

.

“Vague” question Which topological properties does (BX ∗,w∗) enjoy whenever X ∈ SD ??? Given a set A ⊂ X ∗, we write S1(A) :=

• x∗ ∈ X ∗ : ∃(x∗

n) ⊂ A such that w∗- lim n→∞x∗ n = x∗

. Definition We say that (BX ∗,w∗) is

1

Efremov iff S1(C) = C w ∗ for every convex set C ⊂ BX ∗;

“Vague” question Which topological properties does (BX ∗,w∗) enjoy whenever X ∈ SD ??? Given a set A ⊂ X ∗, we write S1(A) :=

• x∗ ∈ X ∗ : ∃(x∗

n) ⊂ A such that w∗- lim n→∞x∗ n = x∗

. Definition We say that (BX ∗,w∗) is

1

Efremov iff S1(C) = C w ∗ for every convex set C ⊂ BX ∗;

2

convexly sequential iff for every convex set C ⊂ BX ∗ we have S1(C) = C = ⇒ C is w∗-closed;

“Vague” question Which topological properties does (BX ∗,w∗) enjoy whenever X ∈ SD ??? Given a set A ⊂ X ∗, we write S1(A) :=

• x∗ ∈ X ∗ : ∃(x∗

n) ⊂ A such that w∗- lim n→∞x∗ n = x∗

. Definition We say that (BX ∗,w∗) is

1

Efremov iff S1(C) = C w ∗ for every convex set C ⊂ BX ∗;

2

convexly sequential iff for every convex set C ⊂ BX ∗ we have S1(C) = C = ⇒ C is w∗-closed; Efremov convexly sequential Fr´ echet-Urysohn sequential

“Vague” question Which topological properties does (BX ∗,w∗) enjoy whenever X ∈ SD ??? Given a set A ⊂ X ∗, we write S1(A) :=

• x∗ ∈ X ∗ : ∃(x∗

n) ⊂ A such that w∗- lim n→∞x∗ n = x∗

. Definition We say that (BX ∗,w∗) is

1

Efremov iff S1(C) = C w ∗ for every convex set C ⊂ BX ∗;

2

convexly sequential iff for every convex set C ⊂ BX ∗ we have S1(C) = C = ⇒ C is w∗-closed; Efremov convexly sequential Fr´ echet-Urysohn sequential sequentially compact

“Vague” question Which topological properties does (BX ∗,w∗) enjoy whenever X ∈ SD ??? Given a set A ⊂ X ∗, we write S1(A) :=

• x∗ ∈ X ∗ : ∃(x∗

n) ⊂ A such that w∗- lim n→∞x∗ n = x∗

. Definition We say that (BX ∗,w∗) is

1

Efremov iff S1(C) = C w ∗ for every convex set C ⊂ BX ∗;

2

convexly sequential iff for every convex set C ⊂ BX ∗ we have S1(C) = C = ⇒ C is w∗-closed;

3

convex block compact iff every sequence in BX ∗ admits a w∗-convergent convex block subsequence. Efremov convexly sequential Fr´ echet-Urysohn sequential sequentially compact convex block compact

“Vague” question Which topological properties does (BX ∗,w∗) enjoy whenever X ∈ SD ??? Given a set A ⊂ X ∗, we write S1(A) :=

• x∗ ∈ X ∗ : ∃(x∗

n) ⊂ A such that w∗- lim n→∞x∗ n = x∗

. Definition We say that (BX ∗,w∗) is

1

Efremov iff S1(C) = C w ∗ for every convex set C ⊂ BX ∗;

2

convexly sequential iff for every convex set C ⊂ BX ∗ we have S1(C) = C = ⇒ C is w∗-closed;

3

convex block compact iff every sequence in BX ∗ admits a w∗-convergent convex block subsequence. Efremov convexly sequential Fr´ echet-Urysohn sequential sequentially compact convex block compact

(Mart´ ınez-Cervantes)

Efremov convexly sequential convex block compact Fr´ echet-Urysohn sequential sequentially compact

Efremov convexly sequential convex block compact Fr´ echet-Urysohn sequential sequentially compact countable tightness

Efremov convexly sequential convex block compact Fr´ echet-Urysohn sequential sequentially compact countable tightness Example Let X = JL2(F) be the Johnson-Lindenstrauss space associated to a MAD family F. Then X ∈ SD

Efremov convexly sequential convex block compact Fr´ echet-Urysohn sequential sequentially compact countable tightness Example Let X = JL2(F) be the Johnson-Lindenstrauss space associated to a MAD family F. Then X ∈ SD and (BX ∗,w∗) is not Fr´ echet-Urysohn.

Efremov convexly sequential convex block compact Fr´ echet-Urysohn sequential sequentially compact countable tightness Example Let X = JL2(F) be the Johnson-Lindenstrauss space associated to a MAD family F. Then X ∈ SD and (BX ∗,w∗) is not Fr´ echet-Urysohn. Moreover: (BX ∗,w∗) is sequential (Mart´ ınez-Cervantes);

Efremov convexly sequential convex block compact Fr´ echet-Urysohn sequential sequentially compact countable tightness Example Let X = JL2(F) be the Johnson-Lindenstrauss space associated to a MAD family F. Then X ∈ SD and (BX ∗,w∗) is not Fr´ echet-Urysohn. Moreover: (BX ∗,w∗) is sequential (Mart´ ınez-Cervantes); Under CH, there exist MAD families F for which (BX ∗,w∗) is/isn’t Efremov (Avil´ es, Mart´ ınez-Cervantes, R.).

Efremov convexly sequential convex block compact Fr´ echet-Urysohn sequential sequentially compact countable tightness Example Let X = JL2(F) be the Johnson-Lindenstrauss space associated to a MAD family F. Then X ∈ SD and (BX ∗,w∗) is not Fr´ echet-Urysohn. Moreover: (BX ∗,w∗) is sequential (Mart´ ınez-Cervantes); Under CH, there exist MAD families F for which (BX ∗,w∗) is/isn’t Efremov (Avil´ es, Mart´ ınez-Cervantes, R.). Theorem If X ∈ SD, then (BX ∗,w∗) has countable tightness [Hern´ andez-Rubio]

Efremov convexly sequential convex block compact Fr´ echet-Urysohn sequential sequentially compact countable tightness Example Let X = JL2(F) be the Johnson-Lindenstrauss space associated to a MAD family F. Then X ∈ SD and (BX ∗,w∗) is not Fr´ echet-Urysohn. Moreover: (BX ∗,w∗) is sequential (Mart´ ınez-Cervantes); Under CH, there exist MAD families F for which (BX ∗,w∗) is/isn’t Efremov (Avil´ es, Mart´ ınez-Cervantes, R.). Theorem If X ∈ SD, then (BX ∗,w∗) has countable tightness [Hern´ andez-Rubio] and is convexly sequential [Avil´ es, Mart´ ınez-Cervantes, R.].

Efremov convexly sequential convex block compact Fr´ echet-Urysohn sequential sequentially compact countable tightness Example Let X = JL2(F) be the Johnson-Lindenstrauss space associated to a MAD family F. Then X ∈ SD and (BX ∗,w∗) is not Fr´ echet-Urysohn. Moreover: (BX ∗,w∗) is sequential (Mart´ ınez-Cervantes); Under CH, there exist MAD families F for which (BX ∗,w∗) is/isn’t Efremov (Avil´ es, Mart´ ınez-Cervantes, R.). Theorem If X ∈ SD, then (BX ∗,w∗) has countable tightness [Hern´ andez-Rubio] and is convexly sequential [Avil´ es, Mart´ ınez-Cervantes, R.]. Question X ∈ SD = ⇒ (BX ∗,w∗) is sequential or sequentially compact ???

X ∈ SD = ⇒ (BX ∗,w∗) is convexly sequential. Ingredients:

X ∈ SD = ⇒ (BX ∗,w∗) is convexly sequential. Ingredients: Theorem (Bourgain) X ⊃ ℓ1 = ⇒ (BX ∗,w∗) is convex block compact.

X ∈ SD = ⇒ (BX ∗,w∗) is convexly sequential. Ingredients: Theorem (Bourgain) X ⊃ ℓ1 = ⇒ (BX ∗,w∗) is convex block compact. Definition Let K ⊂ X ∗ be convex w∗-compact. A set B ⊂ K is a boundary of K iff ∀x ∈ X ∃x∗

0 ∈ B

such that x∗

0(x) = sup{x∗(x) : x∗ ∈ K}.

X ∈ SD = ⇒ (BX ∗,w∗) is convexly sequential. Ingredients: Theorem (Bourgain) X ⊃ ℓ1 = ⇒ (BX ∗,w∗) is convex block compact. Definition Let K ⊂ X ∗ be convex w∗-compact. A set B ⊂ K is a boundary of K iff ∀x ∈ X ∃x∗

0 ∈ B

such that x∗

0(x) = sup{x∗(x) : x∗ ∈ K}.

Theorem (Efremov, Godefroy) X ∈ SD = ⇒ K = conv(B)

·

for all K and B as above.

X ∈ SD = ⇒ (BX ∗,w∗) is convexly sequential. Ingredients: Theorem (Bourgain) X ⊃ ℓ1 = ⇒ (BX ∗,w∗) is convex block compact. Definition Let K ⊂ X ∗ be convex w∗-compact. A set B ⊂ K is a boundary of K iff ∀x ∈ X ∃x∗

0 ∈ B

such that x∗

0(x) = sup{x∗(x) : x∗ ∈ K}.

Theorem (Efremov, Godefroy) X ∈ SD = ⇒ K = conv(B)

·

for all K and B as above. Sketch of proof of the implication: Take C ⊂ BX ∗ convex. Claim: C w ∗ = S1(S1(C)). Why?

X ∈ SD = ⇒ (BX ∗,w∗) is convexly sequential. Ingredients: Theorem (Bourgain) X ⊃ ℓ1 = ⇒ (BX ∗,w∗) is convex block compact. Definition Let K ⊂ X ∗ be convex w∗-compact. A set B ⊂ K is a boundary of K iff ∀x ∈ X ∃x∗

0 ∈ B

such that x∗

0(x) = sup{x∗(x) : x∗ ∈ K}.

Theorem (Efremov, Godefroy) X ∈ SD = ⇒ K = conv(B)

·

for all K and B as above. Sketch of proof of the implication: Take C ⊂ BX ∗ convex. Claim: C w ∗ = S1(S1(C)). Why?

• S1(C) is a boundary of C w ∗

.

X ∈ SD = ⇒ (BX ∗,w∗) is convexly sequential. Ingredients: Theorem (Bourgain) X ⊃ ℓ1 = ⇒ (BX ∗,w∗) is convex block compact. Definition Let K ⊂ X ∗ be convex w∗-compact. A set B ⊂ K is a boundary of K iff ∀x ∈ X ∃x∗

0 ∈ B

such that x∗

0(x) = sup{x∗(x) : x∗ ∈ K}.

Theorem (Efremov, Godefroy) X ∈ SD = ⇒ K = conv(B)

·

for all K and B as above. Sketch of proof of the implication: Take C ⊂ BX ∗ convex. Claim: C w ∗ = S1(S1(C)). Why?

• S1(C) is a boundary of C w ∗

.

• C w ∗

= S1(C)

· ⊂ S1(S1(C)).