Stability in Banach Spaces

Many questions in Banach space theory are of the type: Let X be an infinite dimensional Banach space. Let (P) be a property. Does X contain a closed infinite dimensional subspace Y with (P)? Sometimes the question takes the form: If X has a certain property (Q), does X contain Y having (P)? Of course the answers and solutions (if known) depend upon the particular properties involved. But there is perhaps one theme that runs throughout many of these problems. One often wishes to stabilize some function on some substructure of X. This might be achieved quite simply, say by the pigeonhole principle or an analytical argument using compactness or may be more involved using infinitary combinatorics (e. g., Ramsey theory). One may require new ad hoc arguments that in turn lead to new combinatorial results. Before continuing we set some notation and give some background. A Banach space (X, ‖·‖) is a complete normed linear space. A norm ||| · ||| on X is an equivalent norm if for some constants A,B > 0 for all x ∈ X A−1|||x||| ≤ ‖x‖ ≤ B|||x||| . The norm ‖·‖ on X is determined by the unit ball BX ≡ {x ∈ X : ‖x‖ ≤ 1} or by the unit sphere SX = {x ∈ X : ‖x‖ = 1}. ||| · ||| is an equivalent norm just means that B−1B(X,|||·|||) ⊆ BX ⊆ A ·B(X,|||·|||) ∗Research supported by NSF.