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 $dim{H}^q(X,V\otimes E)\<\infty$. In particular, if $dim{H}^q(X,𝒪)=0$, then $dim{H}^q(X,E)\<\infty$.

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.

The inequality $ʃlog|\sum a_n{e}^{2\pi i{\phi }_n}|d{\phi }_1\dots d{\phi }_n\ge Clog\left(\sum \right|a_n{{|}^2)}^{1/2}$ with an absolute constant C, and similar ones, are extended to the case of $a_n$ belonging to an arbitrary normed space X and an arbitrary compact group of unitary operators on X instead of the operators of multiplication by $e^{2\pi i\phi }$.

In this paper, a precise projection decomposition in reflexive, smooth and strictly convex Orlicz-Bochner spaces is given by the representation of the duality mapping. As an application, a representation of the metric projection operator on a closed hyperplane is presented.

The first and last sections of this paper are intended for a general mathematical audience. In addition to some very brief remarks of a somewhat historical nature, we pose a rather simply formulated question in the realm of (discrete) geometry. This question has arisen in connection with a recently developed approach for studying various versions of the function space BMO. We describe that approach and the results that it gives. Special cases of one of our results give alternative proofs of the...

The famous Gowers tree space is the first example of a space not containing c₀, ℓ₁ or a reflexive subspace. We present a space with a similar construction and prove that it is hereditarily indecomposable (HI) and has ℓ₂ as a quotient space. Furthermore, we show that every bounded linear operator on it is of the form λI + W where W is a weakly compact (hence strictly singular) operator.