Large cardinals and basic sequences Large cardinals and basic sequences J. Lopez-Abad Instituto de Ciencias Matem´ aticas. CSIC. Madrid, Spain. LC2011, July 2011
Large cardinals and basic sequences Introduction The intend of this talk is to present a combinatorial approach to the study of the structure of a Banach space. In particular, we will focus on the existence of certain sequences, for example unconditional sequences.
Large cardinals and basic sequences Introduction Problems: (1) The separable quotient problem. (2) Subspaces. (3) Special sequences: basic, unconditional, subsymmetric...
Large cardinals and basic sequences Introduction Tools: (1) Structural Ramsey theorems. (2) (Not so) large cardinal numbers. (3) Forcing.
Large cardinals and basic sequences Basic definitions and results Definition A Banach space ( X , � · � ) is a vector space X (over the real numbers) endowed with a norm � · � (N.1) � λ x � = | λ |� x � (N.2) � x + y � ≤ � x � + � y � (N.3) � x � = 0 iff x = 0 which is complete , i.e. Cauchy sequences are convergent.
Large cardinals and basic sequences Basic definitions and results Examples: • R n with the euclidean norm � ( a i ) i < n � 2 = ( � 1 i < n | a i | 2 ) 2 , • the infinite dimensional separable Hilbert space 1 i ∈ N | a i | 2 ) ℓ 2 = { ( a i ) i ∈ N : ( � 2 < ∞} , with the euclidean norm 1 i ∈ N | a i | 2 ) 2 , � ( a i ) i ∈ N � 2 = ( � 1 i ∈ N | a i | p ) p < ∞} , • the ℓ p spaces, for p ≥ 1, ℓ p = { ( a i ) i ∈ N : ( � 1 p , i ∈ N | a i | p ) with the p -norm � ( a i ) i ∈ N � p = ( �
Large cardinals and basic sequences Basic definitions and results • c 0 = { ( a i ) i ∈ N : lim i →∞ a i = 0 } , with the sup-norm � ( a i ) i ∈ N � ∞ = sup {| a i | : i ∈ N } , • ℓ ∞ = { ( a i ) i ∈ N : sup i →∞ | a i | < ∞} , with the sup-norm � ( a i ) i ∈ N � ∞ = sup {| a i | : i ∈ N } , • for a compact space K , the space C ( K ) of continuous functions on K , endowed with the sup-norm, � f � = sup {| f ( x ) | : x ∈ K } . In particular, C ([ 0 , 1 ]) .
Large cardinals and basic sequences Basic definitions and results While ( R n , � · � p ) are all isomorphic, the infinite dimensional versions are very much different:
Large cardinals and basic sequences Basic definitions and results Basic notions • A Banach space is infinite dimensional if it is not finite dimensional. • The density of a space X is the topological weight, i.e. the smallest cardinality of a dense subset of X . • A subspace Y of X will be understood as a linear subspace of X , which is in addition closed . In particular Y with the norm � · � is also a Banach space. • Given a subspace Y of X , the quotient space X / Y is the Banach space over the linear quotient, endowed with the norm � x + Y � := d ( x , Y ) .
Large cardinals and basic sequences Basic definitions and results • An operator T : X → Y between two spaces X and Y is a linear mapping which is continuous, or equivalently bounded , i.e., such that � T � := sup {� Tx � : x ∈ X , � x � ≤ 1 } < ∞ . • An isomorphic embedding T : X → Y is a 1-1 operator such that T ( X ) is a closed subspace of Y and the inverse U : T ( X ) → X is bounded. • The dual X ∗ of a Banach space X the space of all operator f : X → R . This is a Banach space with the norm � f � := sup {� f ( x ) � : � x � ≤ 1 } . The elements of X ∗ are called functionals . • A space X is called reflexive when X ∗∗ is canonically identified with X .
Large cardinals and basic sequences Basic definitions and results Special sequences • A sequence ( x γ ) γ<κ in a Banach space ( X , � · � ) , indexed in some cardinal number κ is called a biorthogonal sequence if for every α < κ there is a functional f α ∈ X ∗ such that f α ( x β ) = δ α,β . Notice that in particular biorthogonal sequences are linearly independent sequences. • A normalized sequence ( x γ ) γ<κ in a Banach space ( X , � · � ) , indexed in some cardinal number κ is called a (Schauder) basic sequence when there is a constant C ≥ 1 such that � � � a γ x γ � ≤ C � a γ x γ � γ ∈ s γ ∈ t for every sequence of scalars ( a γ ) γ ∈ s and every t ⊑ s ⊆ κ . It follows easily that ( x α ) α is a biorthogonal sequence.
Large cardinals and basic sequences Basic definitions and results • A normalized sequence ( x γ ) γ<κ in X is a (Schauder) basis of X if it is basic, and the closed linear span of ( x α ) α is X . Equivalently, every x ∈ X has a unique representation as � x = a γ x γ . γ<κ
Large cardinals and basic sequences Basic definitions and results • ( x γ ) γ is called an unconditional basic sequence when there is a constant C ≥ 1 such that C ≥ 1 such that � � � a γ x γ � ≤ C � a γ x γ � γ ∈ s γ ∈ t for every sequence of scalars ( a γ ) γ ∈ s and every t ⊆ s ⊆ κ . • ( x γ ) γ is called subsymmetric when there is a constant C ≥ 1 such that � � � a k x γ k � ≤ C � a k x ξ k � k ∈ l k < l for every sequence of scalars ( a k ) k < l and every γ 0 < · · · < γ l − 1 , ξ 0 < · · · < ξ l − 1 .
Large cardinals and basic sequences Basic definitions and results • ( x γ ) γ is called weakly-null when the set { γ < κ : | x ∗ ( x γ ) | ≥ ε } is finite for every x ∗ ∈ X ∗ and every ε > 0.
Large cardinals and basic sequences Basic definitions and results Examples: ( n ) The unit basis ( u n ) n , u n = ( 0 , 0 , ..., 0 , 1 , 0 , 0 , ... ) , is a Schauder basis of each ℓ p or c 0 . Indeed it is an unconditional and subsymmetric basis. For p > 1 it is also weakly-null.
Large cardinals and basic sequences Basic definitions and results Basic results about special sequences • ( Banach-Mazur ) Every infinite dimensional Banach space has a basic sequence. • ( Mazur ) Every normalized weakly-null sequence has a basic subsequence. • ( Enflo ) There are separable Banach spaces without bases. • ( Bessaga-Pelczynski ) The structure of subspaces of a Banach space with a basis ( e n ) n is determined by block subsequences of the basis ( e n ) n . • ( James ) The reflexivity of a space with a basis is determined by the basis. • ( James ) A space with an unconditional basis is reflexive iff it does not contain an isomorphic copy of c 0 or ℓ 1 . • ( Krivine ) c 0 or ℓ p are finitely block representable in any Banach space.
Large cardinals and basic sequences Basic definitions and results • ( Tsirelson ) There is a space T without isomorphic copies of c 0 or ℓ p , p ≥ 1. Indeed T does not have subsymmetric basic sequences. ( n + 1 ) • The summing basis s n = ( 1 , ..., 1 , 0 , 0 , . . . ) ∈ c 0 does not have unconditional subsequences, and it is not weakly-null. • ( Rosenthal ) Every norm-bounded sequence ( x n ) n has a subsequence which is either equivalent to the unit basis of ℓ 1 or a weakly-Cauchy, i.e., ( f ( x n )) n is a numerical Cauchy sequence for each f ∈ X ∗ . • ( Maurey-Rosenthal ) There is a weakly-null basic sequence without unconditional subsequences. • ( Gowers-Maurey ) There is a reflexive Banach space without unconditional basic sequences.
Large cardinals and basic sequences The problems: Uncountable sequences Main goal: Existence of uncountable special sequences on a given Banach space. Theorem (Amir-Lindenstrauss) Every reflexive space of infinite density κ has a normalized weakly-null sequence of length κ . Question Is it true that a non-separable space has an uncountable biorthogonal sequence? Question Is it true that a non-separable space has an uncountable basic sequence?
Large cardinals and basic sequences The problems: Uncountable sequences Theorem (Kunen) CH implies that there is a non-metrizable scattered compactum K such that C ( K ) does not have uncountable biorthogonal sequences. Theorem (Todorcevic) b = ω 1 implies that there is a non-metrizable scattered compactum K such that C ( K ) does not have uncountable biorthogonal sequences.
Large cardinals and basic sequences The problems: Uncountable sequences Theorem (Shelah) the ♦ principle implies that there is a non-separable Gurarij space without uncountable biorthogonal sequences. Theorem (Johnson-Lindenstrauss) There is a non-separable space without uncountable basic sequences.
Large cardinals and basic sequences The problems: Uncountable sequences By the method of forcing it is possible to build generic spaces as direct limits of finite dimensional polyhedral spaces and isometries between them, providing a large variety of examples of spaces having certain nice uncountable sequences and not others, as well as nice geometrical properties. Applications: 1 Lindenstrauss spaces (preduals of L 1 ) 2 Gurarij spaces 3 Distinction between different uncountable substructures of the generic Banach space (e.g. biorthogonal-like systems).
Recommend
More recommend
