An ideal boundary for domains in -space.
C.-M. Cho and W. B. Johnson showed that if a subspace E of , 1 < p < ∞, has the compact approximation property, then K(E) is an M-ideal in ℒ(E). We prove that for every r,s ∈ ]0,1] with , the James space can be provided with an equivalent norm such that an arbitrary subspace E has the metric compact approximation property iff there is a norm one projection P on ℒ(E)* with Ker P = K(E)⊥ satisfying ∥⨍∥ ≥ r∥Pf∥ + s∥φ - Pf∥ ∀⨍ ∈ ℒ(E)*. A similar result is proved for subspaces of upper p-spaces...
We construct an indecomposable reflexive Banach space such that every infinite-dimensional closed subspace contains an unconditional basic sequence. We also show that every operator is of the form λI + S with S a strictly singular operator.
Using the method of forcing we construct a model for ZFC where CH does not hold and where there exists a connected compact topological space K of weight such that every operator on the Banach space of continuous functions on K is multiplication by a continuous function plus a weakly compact operator. In particular, the Banach space of continuous functions on K is indecomposable.
We derive a formula for the index of Fredholm chains on normed spaces.
We consider the James and Schäffer type constants recently introduced by Takahashi. We prove an equality between James (resp. Schäffer) type constants and the modulus of convexity (resp. smoothness). By using these equalities, we obtain some estimates for the new constants in terms of the James constant. As a result, we improve an inequality between the Zbăganu and James constants.
We use Simonenko quantitative indices of an -function to estimate two parameters and in Orlicz function spaces with Orlicz norm, and get the following inequality: , where and are Simonenko indices. A similar inequality is obtained in with Orlicz norm.