A family is constructed of cardinality equal to the continuum, whose members are totally incomparable hereditarily indecomposable Banach spaces.

We prove the existence of a contractive mapping on a weakly compact convex set in a Banach space that is fixed point free. This answers a long-standing open question.

We show that a complex Banach space is weakly Lindelöf determined if and only if the dual unit ball of any equivalent norm is weak* Valdivia compactum. We deduce that a complex Banach space X is weakly Lindelöf determined if and only if any nonseparable Banach space isomorphic to a complemented subspace of X admits a projectional resolution of the identity. These results complete the previous ones on real spaces.

Let Y be a Banach space, (Ω, Σ; μ) a probability space and φ a finite Young function. It is shown that the Y-valued Orlicz heart H φ(μ, Y) is isometrically isomorphic to the l-completed tensor product $$H_{\varphi }\left(\mu \right){\tilde{\otimes }}_{l}Y$$ of the scalar-valued Orlicz heart Hφ(μ) and Y, in the sense of Chaney and Schaefer. As an application, a characterization is given of the equality of $$\left(H_{\varphi }\left(\mu \right){\tilde{\otimes }}_{l}Y\right)*$$ and $$H_{\varphi }\left(\mu \right)*{\tilde{\otimes }}_{l}Y*$$ in terms of the Radon-Nikodým property on Y. Convergence of norm-bounded martingales in H φ(μ, Y) is characterized in terms of the Radon-Nikodým...

We show that the Schreier sets $S_{\alpha }(\alpha <{\omega }_1)$ have the following dichotomy property. For every hereditary collection ℱ of finite subsets of ℱ, either there exists infinite $M={\left(m_i\right)}_{i=1}^{\infty }\subseteq \mathbb{N}$ such that $S_{\alpha }\left(M\right)=m_i:i\in E:E\in S_{\alpha }\subseteq ℱ$, or there exist infinite $M={\left(m_i\right)}_{i=1}^{\infty },N\subseteq \mathbb{N}$ such that $ℱ\left[N\right]\left(M\right)=m_i:i\in F:F\in ℱandF\subset N\subseteq S_{\alpha }$.

Let p ∈ (1,∞). The question of existence of a curve in ℝ₊² starting at (0,0) and such that at every point (x,y) of this curve, the $l^p$-distance of the points (x,y) and (0,0) is equal to the Euclidean length of the arc of this curve between these points is considered. This problem reduces to a nonlinear differential equation. The existence and uniqueness of solutions is proved and nonelementary explicit solutions are given.

We give an alternative proof of W. T. Gowers' theorem on block bases by reducing it to a discrete analogue on specific countable nets. We also give a Ramsey type result on k-tuples of block sequences in a normed linear space with a Schauder basis.

We give a characterization of conditional expectation operators through a disjointness type property similar to band-preserving operators. We say that the operator T:X→ X on a Banach lattice X is semi-band-preserving if and only if for all f, g ∈ X, f ⊥ Tg implies that Tf ⊥ Tg. We prove that when X is a purely atomic Banach lattice, then an operator T on X is a weighted conditional expectation operator if and only if T is semi-band-preserving.

For Banach lattices X with strictly or uniformly monotone lattice norm dual, properties (o)-smoothness and (o)-uniform smoothness are introduced. Lindenstrauss type duality formulas are proved and duality theorems are derived. It is observed that (o)-uniformly smooth Banach lattices X are order dense in X**. An application to an optimization theorem is given.

Let $E$ be a holomorphic Banach bundle over a compact complex manifold, which can be defined by a cocycle of holomorphic transition functions with values of the form $\mathrm{id}+K$ where $K$ is compact. Assume that the characteristic fiber of $E$ has the compact approximation property. Let $n$ be the complex dimension of $X$ and $0\le q\le n$. Then: If $V\to X$ is a holomorphic vector bundle (of finite rank) with $H^q(X,V)=0$, then $dim{H}^q(X,V\otimes E)\&lt;\infty$. In particular, if $dim{H}^q(X,𝒪)=0$, then $dim{H}^q(X,E)\&lt;\infty$.

For a holomorphic function ψ defined on a sector we give a condition implying the identity ${(X,\left(A^{\alpha }\right))}_{\theta ,p}=x\in X|t^{-\theta Re\alpha }\psi \left(tA\right)\in L{⁎}^p((0,\infty );X)$ where A is a sectorial operator on a Banach space X. This yields all common descriptions of the real interpolation spaces for sectorial operators and allows easy proofs of the moment inequalities and reiteration results for fractional powers.