It is proved that the Musielak-Orlicz function space LF(mu,X) of Bochner type is P-convex if and only if both spaces LF(mu,R) and X are P-convex. In particular, the Lebesgue-Bochner space Lp(mu,X) is P-convex iff X is P-convex.

We prove a very general theorem concerning the estimation of the expression ||T((a+b)/2) - (Ta+Tb)/2|| for different kinds of maps T satisfying some general perturbed isometry condition. It can be seen as a quantitative generalization of the classical Mazur-Ulam theorem. The estimates improve the existing ones for bi-Lipschitz maps. As a consequence we also obtain a very simple proof of the result of Gevirtz which answers the Hyers-Ulam problem and we prove a non-linear generalization of the Banach-Stone...

We describe the geometrical structure on a complex quasi-Banach space $X$ that is necessay and sufficient for the existence of boundary limits for bounded, $X$-valued analytic functions on the open unit disc of the complex plane. It is shown that in such spaces, closed bounded subsets have many plurisubharmonic barriers and that bounded upper semi-continuous functions on these sets have arbitrarily small plurisubharmonic perturbations that attain their maximum. This yields a certain representation of...

We prove that for each linear contraction T : X → X (∥T∥ ≤ 1), the subspace F = {x ∈ X : Tx = x} of fixed points is 1-complemented, where X is a suitable subspace of L¹(E*) and E* is a separable dual space such that the weak and weak* topologies coincide on the unit sphere. We also prove some related fixed point results.

For $U$ a balanced open subset of a Fréchet space $E$ and $F$ a dual-Banach space we introduce the topology ${\tau }_{\gamma }$ on the space $\mathscr{H}(U,F)$ of holomorphic functions from $U$ into $F$. This topology allows us to construct a predual for $(\mathscr{H}(U,F),{\tau }_{\delta })$ which in turn allows us to investigate the topological structure of spaces of vector-valued holomorphic functions. In particular, we are able to give necessary and sufficient conditions for the equivalence and compatibility of various topologies on spaces of vector-valued holomorphic functions....

Nous montrons que $h^2(𝔻,L^1\left(𝕋\right))$ admet une norme équivalente $LUR,$ ce qui répond négativement à une question de Dowling, Hu et Smith. Puis nous obtenons une propriété de stabilité des opérateurs de Radon-Nikodym analytique. Motivés par l’identification entre $h^p(𝔻,X)$ et $V{B}^p(𝕋,X)$ où $X$ est un espace de Banach, pour un groupe abélien compact métrisable $G$, son dual $\Gamma$, et ${\Lambda }_2\subset {\Lambda }_1\subset \Gamma$, nous prouvons que, si l’espace $V{B}_{{\Lambda }_1}^p(G,X)/V{B}_{{\Lambda }_2}^p(G,X)$ a la propriété $Kadec-Klee-{\beta }^{\prime }-\omega$, alors il coincïde avec $L_{{\Lambda }_1}^p(G,X)/L_{{\Lambda }_2}^p(G,X),$$...$