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´ on/FEDER (MTM2017-86182-P) and Fundaci´ on S´ eneca (20797/PI/18)

Throughout this talk X is a Banach space.

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

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

Throughout this talk X is a Banach space. Goldstine’s theorem B X is w ∗ -dense in B X ∗∗ . 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 B X is w ∗ -dense in B X ∗∗ . 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 B X is w ∗ -dense in B X ∗∗ . 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 ⇒ ( B X ∗∗ , w ∗ ) is metrizable ] [ ⇐

Throughout this talk X is a Banach space. Goldstine’s theorem B X is w ∗ -dense in B X ∗∗ . 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 ⇒ ( B X ∗∗ , w ∗ ) is metrizable ] [ ⇐ ( B X ∗∗ , w ∗ ) is Fr´ echet-Urysohn = ⇒ X ∈ SD

Throughout this talk X is a Banach space. Goldstine’s theorem B X is w ∗ -dense in B X ∗∗ . Therefore, X is w ∗ -dense in X ∗∗ . Notation X ∈ SD iff X is w ∗ - sequentially dense in X ∗∗ . X ∈ SD if X �∈ SD if X is reflexive (obvious) X is weakly sequentially complete X ∗ is separable and non-reflexive (e.g. ℓ 1 and L 1 ) ⇒ ( B X ∗∗ , w ∗ ) is metrizable ] [ ⇐ ( B X ∗∗ , w ∗ ) is Fr´ echet-Urysohn = ⇒ X ∈ SD

Throughout this talk X is a Banach space. Goldstine’s theorem B X is w ∗ -dense in B X ∗∗ . Therefore, X is w ∗ -dense in X ∗∗ . Notation X ∈ SD iff X is w ∗ - sequentially dense in X ∗∗ . X ∈ SD if X �∈ SD if X is reflexive (obvious) X is weakly sequentially complete X ∗ is separable and non-reflexive (e.g. ℓ 1 and L 1 ) ⇒ ( B X ∗∗ , w ∗ ) is metrizable ] X = c 0 (Γ) for uncountable Γ [ ⇐ ( B X ∗∗ , w ∗ ) is Fr´ echet-Urysohn = ⇒ X ∈ SD

Throughout this talk X is a Banach space. Goldstine’s theorem B X is w ∗ -dense in B X ∗∗ . Therefore, X is w ∗ -dense in X ∗∗ . Notation X ∈ SD iff X is w ∗ - sequentially dense in X ∗∗ . X ∈ SD if X �∈ SD if X is reflexive (obvious) X is weakly sequentially complete X ∗ is separable and non-reflexive (e.g. ℓ 1 and L 1 ) ⇒ ( B X ∗∗ , w ∗ ) is metrizable ] X = c 0 (Γ) for uncountable Γ [ ⇐ ( B X ∗∗ , w ∗ ) is Fr´ echet-Urysohn = ⇒ X ∈ SD = ⇒ X �⊃ ℓ 1 .

Throughout this talk X is a Banach space. Goldstine’s theorem B X is w ∗ -dense in B X ∗∗ . Therefore, X is w ∗ -dense in X ∗∗ . Notation X ∈ SD iff X is w ∗ - sequentially dense in X ∗∗ . X ∈ SD if X �∈ SD if X is reflexive (obvious) X is weakly sequentially complete X ∗ is separable and non-reflexive (e.g. ℓ 1 and L 1 ) ⇒ ( B X ∗∗ , w ∗ ) is metrizable ] X = c 0 (Γ) for uncountable Γ [ ⇐ ( B X ∗∗ , w ∗ ) is Fr´ echet-Urysohn = ⇒ X ∈ SD = ⇒ X �⊃ ℓ 1 . Theorem (Odell-Rosenthal, Bourgain-Fremlin-Talagrand) Suppose X is separable . Then: ( B X ∗∗ , w ∗ ) is Fr´ echet-Urysohn ⇐ ⇒ X ∈ SD ⇐ ⇒ X �⊃ ℓ 1 .

( B X ∗∗ , w ∗ ) is Fr´ echet-Urysohn X ∈ SD X �⊃ ℓ 1

( B X ∗∗ , w ∗ ) is Fr´ echet-Urysohn X ∈ SD X �⊃ ℓ 1 ( B X ∗∗ , w ∗ ) is Corson

( B X ∗∗ , w ∗ ) is Fr´ echet-Urysohn X ∈ SD X �⊃ ℓ 1 X ∗ is WCG ( B X ∗∗ , w ∗ ) is Corson

( B X ∗∗ , w ∗ ) is Fr´ echet-Urysohn X ∈ SD X �⊃ ℓ 1 X ∗ is WCG ( B X ∗∗ , w ∗ ) is Corson X is Asplund

( B X ∗∗ , w ∗ ) is Fr´ echet-Urysohn X ∈ SD X �⊃ ℓ 1 X ∗ is WCG ( B X ∗∗ , w ∗ ) is Corson X is Asplund Theorem (Deville-Godefroy, Orihuela) ( B X ∗∗ , w ∗ ) is Corson ⇐ ⇒ X ∈ SD and X is Asplund.

( B X ∗∗ , w ∗ ) is Fr´ echet-Urysohn X ∈ SD X �⊃ ℓ 1 X ∗ is WCG ( B X ∗∗ , w ∗ ) is Corson X is Asplund Theorem (Deville-Godefroy, Orihuela) ( B X ∗∗ , w ∗ ) is Corson ⇐ ⇒ X ∈ SD and X is Asplund. A Banach lattice X is Asplund if and only if X �⊃ ℓ 1 .

( B X ∗∗ , w ∗ ) is Fr´ echet-Urysohn X ∈ SD X �⊃ ℓ 1 X ∗ is WCG ( B X ∗∗ , w ∗ ) is Corson X is Asplund Theorem (Deville-Godefroy, Orihuela) ( B X ∗∗ , w ∗ ) is Corson ⇐ ⇒ X ∈ SD and X is Asplund. A Banach lattice X is Asplund if and only if X �⊃ ℓ 1 . Corollary Suppose X is a Banach lattice . Then ( B X ∗∗ , w ∗ ) is Corson ⇐ ⇒ X ∈ SD .

( B X ∗∗ , w ∗ ) is Fr´ echet-Urysohn X ∈ SD X �⊃ ℓ 1 X ∗ is WCG ( B X ∗∗ , w ∗ ) is Corson X is Asplund Theorem (Deville-Godefroy, Orihuela) ( B X ∗∗ , w ∗ ) is Corson ⇐ ⇒ X ∈ SD and X is Asplund. A Banach lattice X is Asplund if and only if X �⊃ ℓ 1 . Corollary Suppose X is a Banach lattice . Then ( B X ∗∗ , w ∗ ) is Corson ⇐ ⇒ X ∈ SD . Question ⇒ ( B X ∗∗ , w ∗ ) is Fr´ X ∈ SD = echet-Urysohn ???

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

“Vague” question Which topological properties does ( B X ∗ , w ∗ ) enjoy whenever X ∈ SD ??? Given a set A ⊂ X ∗ , we write x ∗ ∈ X ∗ : ∃ ( x ∗ n ) ⊂ A such that w ∗ - lim n → ∞ x ∗ n = x ∗ � � S 1 ( A ) := .

“Vague” question Which topological properties does ( B X ∗ , w ∗ ) enjoy whenever X ∈ SD ??? Given a set A ⊂ X ∗ , we write x ∗ ∈ X ∗ : ∃ ( x ∗ n ) ⊂ A such that w ∗ - lim n → ∞ x ∗ n = x ∗ � � S 1 ( A ) := . Definition We say that ( B X ∗ , w ∗ ) is Efremov iff S 1 ( C ) = C w ∗ for every convex set C ⊂ B X ∗ ; 1

“Vague” question Which topological properties does ( B X ∗ , w ∗ ) enjoy whenever X ∈ SD ??? Given a set A ⊂ X ∗ , we write x ∗ ∈ X ∗ : ∃ ( x ∗ n ) ⊂ A such that w ∗ - lim n → ∞ x ∗ n = x ∗ � � S 1 ( A ) := . Definition We say that ( B X ∗ , w ∗ ) is Efremov iff S 1 ( C ) = C w ∗ for every convex set C ⊂ B X ∗ ; 1 convexly sequential iff for every convex set C ⊂ B X ∗ we have 2 C is w ∗ -closed; S 1 ( C ) = C = ⇒

“Vague” question Which topological properties does ( B X ∗ , w ∗ ) enjoy whenever X ∈ SD ??? Given a set A ⊂ X ∗ , we write x ∗ ∈ X ∗ : ∃ ( x ∗ n ) ⊂ A such that w ∗ - lim n → ∞ x ∗ n = x ∗ � � S 1 ( A ) := . Definition We say that ( B X ∗ , w ∗ ) is Efremov iff S 1 ( C ) = C w ∗ for every convex set C ⊂ B X ∗ ; 1 convexly sequential iff for every convex set C ⊂ B X ∗ we have 2 C is w ∗ -closed; S 1 ( C ) = C = ⇒ convexly sequential Efremov Fr´ echet-Urysohn sequential

“Vague” question Which topological properties does ( B X ∗ , w ∗ ) enjoy whenever X ∈ SD ??? Given a set A ⊂ X ∗ , we write x ∗ ∈ X ∗ : ∃ ( x ∗ n ) ⊂ A such that w ∗ - lim n → ∞ x ∗ n = x ∗ � � S 1 ( A ) := . Definition We say that ( B X ∗ , w ∗ ) is Efremov iff S 1 ( C ) = C w ∗ for every convex set C ⊂ B X ∗ ; 1 convexly sequential iff for every convex set C ⊂ B X ∗ we have 2 C is w ∗ -closed; S 1 ( C ) = C = ⇒ convexly sequential Efremov Fr´ echet-Urysohn sequential sequentially compact

“Vague” question Which topological properties does ( B X ∗ , w ∗ ) enjoy whenever X ∈ SD ??? Given a set A ⊂ X ∗ , we write x ∗ ∈ X ∗ : ∃ ( x ∗ n ) ⊂ A such that w ∗ - lim n → ∞ x ∗ n = x ∗ � � S 1 ( A ) := . Definition We say that ( B X ∗ , w ∗ ) is Efremov iff S 1 ( C ) = C w ∗ for every convex set C ⊂ B X ∗ ; 1 convexly sequential iff for every convex set C ⊂ B X ∗ we have 2 C is w ∗ -closed; S 1 ( C ) = C = ⇒ convex block compact iff every sequence in B X ∗ admits a w ∗ -convergent 3 convex block subsequence. convexly sequential convex block compact Efremov Fr´ echet-Urysohn sequential sequentially compact

“Vague” question Which topological properties does ( B X ∗ , w ∗ ) enjoy whenever X ∈ SD ??? Given a set A ⊂ X ∗ , we write x ∗ ∈ X ∗ : ∃ ( x ∗ n ) ⊂ A such that w ∗ - lim n → ∞ x ∗ n = x ∗ � � S 1 ( A ) := . Definition We say that ( B X ∗ , w ∗ ) is Efremov iff S 1 ( C ) = C w ∗ for every convex set C ⊂ B X ∗ ; 1 convexly sequential iff for every convex set C ⊂ B X ∗ we have 2 C is w ∗ -closed; S 1 ( C ) = C = ⇒ convex block compact iff every sequence in B X ∗ admits a w ∗ -convergent 3 convex block subsequence. (Mart´ ınez-Cervantes) convexly sequential convex block compact Efremov Fr´ echet-Urysohn sequential sequentially compact