We define the class of integral holomorphic functions over Banach spaces; these are functions admitting an integral representation akin to the Cauchy integral formula, and are related to integral polynomials. After studying various properties of these functions, Banach and Fréchet spaces of integral holomorphic functions are defined, and several aspects investigated: duality, Taylor series approximation, biduality and reflexivity.

Let $\theta \in (0,1)$, $\lambda \in [0,1)$ and $p,p_0,p_1\in (1,\infty ]$ be such that $(1-\theta )/p_0+\theta /p_1=1/p$, and let $\varphi ,{\varphi }_0,{\varphi }_1$ be some admissible functions such that $\varphi ,{\varphi }_0^{p/p_0}$ and ${\varphi }_1^{p/p_1}$ are equivalent. We first prove that, via the $\pm$ interpolation method, the interpolation $\langle L_{{\varphi }_0}^{p_0),\lambda }\left(𝒳\right),L_{{\varphi }_1}^{p_1),\lambda }\left(𝒳\right),\theta \rangle$ of two generalized grand Morrey spaces on a quasi-metric measure space $𝒳$ is the generalized grand Morrey space $L_{\varphi }^{p),\lambda }\left(𝒳\right)$. Then, by using block functions, we also find a predual space of the generalized grand Morrey space. These results are new even for generalized grand Lebesgue spaces.

We study conditions on an infinite dimensional separable Banach space $X$ implying that $X$ is the only non-trivial invariant subspace of $X^{**}$ under the action of the algebra $𝔸\left(X\right)$ of biconjugates of bounded operators on $X$: $𝔸\left(X\right)=\{T^{**}T\in ℬ\left(X\right)\}$. Such a space is called simple. We characterize simple spaces among spaces which contain an isomorphic copy of $c_0$, and show in particular that any space which does not contain ${\ell }_1$ and has property (u) of Pelczynski is simple.

We study the local dual spaces of a Banach space X, which can be described as the subspaces of X* that have the properties that the principle of local reflexivity attributes to X as a subspace of X**. We give several characterizations of local dual spaces, which allow us to show many examples. Moreover, every separable space X has a separable local dual Z, and we can choose Z with the metric approximation property if X has it. We also show that a separable space containing no...

The notion of functions dependent locally on finitely many coordinates plays an important role in the theory of smoothness and renormings on Banach spaces, especially when higher order smoothness is involved. In this note we investigate the structural properties of Banach spaces admitting (arbitrary) bump functions depending locally on finitely many coordinates.

A metric space (M,d) is said to have the small ball property (sbp) if for every ε₀ > 0 it is possible to write M as the union of a sequence (B(xₙ,rₙ)) of closed balls such that the rₙ are smaller than ε₀ and lim rₙ = 0. We study permanence properties and examples of sbp. The main results of this paper are the following: 1. Bounded convex closed sets in Banach spaces have sbp only if they are compact. 2. Precisely the finite-dimensional Banach spaces have sbp. (More generally: a complete metric...

We study Banach space properties of non-commutative martingale VMO-spaces associated with general von Neumann algebras. More precisely, we obtain a version of the classical Kadets-Pełczyński dichotomy theorem for subspaces of non-commutative martingale VMO-spaces. As application we prove that if ℳ is hyperfinite then the non-commutative martingale VMO-space associated with a filtration of finite-dimensional von Neumannn subalgebras of ℳ has property (u).