On some properties of Banach spaces. (Spanish: Sobre algunas propiedades de espacios de Banach)

Abstract

This paper is devoted to an exposition of the two main methods used to introduce isomorphic properties of Banach spaces. The first one consists in the comparison of different classes of operators; the second in the comparison of different classes of subsets. For instance, it is well known that a Banach space E is finite-dimensional if and only if every bounded linear operator on E is compact if and only if every closed, bounded subset of E is compact. The first method, introduced by Grothendieck, is based upon the following general scheme: Let Φ,Ψ be two classes of bounded linear operators between Banach spaces and E a class of Banach spaces. Then E is said to have property P(Φ,Ψ,E) if Φ(E,F)⊆Ψ(E,F) for all F∈E ; if E is the class of all Banach spaces one simply writes E∈P(Φ,Ψ) . The author considers the following classes of operators: K(E,F)= compact operators; W(E,F)= weakly compact operators; DP(E,F)= Dunford-Pettis operators; D(E,F)= Dieudonné operators; UC(E,F)= unconditionally convergent operators; L(E,F)= bounded linear operators. Then he gives the following definition: A Banach space E (i) has the Dunford-Pettis property if E∈P(W,DP) ; (ii) has the reciprocal Dunford-Pettis property if E∈P(DP,W) ; (iii) has the Dieudonné property if E∈P(D,W) ; (iv) has property (V) if E∈P(UC,W) ; (v) has the Schur property if E∈P(L,DP) ; (vi) is weakly sequentially complete if E∈P(L,D) ; (vii) does not contain copies of c 0 if E∈P(L,UC) ; (viii) does not contain copies of l 1 if E∈P(DP,K) ; (ix) has the Grothendieck property if E∈P(L,W,c 0 ) . The author also shows that the above properties are always inherited by finite products and complemented subspaces, whereas quotients or closed subspaces inherit them just in special cases. Then the author treats the second method of introducing isomorphic properties, by considering several classes of subsets of a Banach space E , namely: K(E)= relatively compact subsets; W(E)= weakly relatively compact subsets; WC(E)= weakly conditionally compact subsets; B(E)= bounded subsets; L ∗ (E)= limited subsets; DP(E)= Dunford-Pettis subsets; V ∗ (E)=V ∗ subsets. He shows that the inclusion of one of them into another is useful for giving a different formulation of the above definition (example: E has the Dunford-Pettis property if W(E)⊆DP(E) ) as well as for introducing new properties (example: E is said to have the Gelʹfand-Phillips property if L ∗ (E)⊆K(E) ). Several results about this second method are then presented.