Valov proved a general version of Arvanitakis's simultaneous selection theorem which is a common generalization of both Michael's selection theorem and Dugundji's extension theorem. We show that Valov's theorem can be extended by applying an argument by means of Pettis integrals due to Repovš, Semenov and Shchepin.

Define $h^{\infty }\left(E\right)$ as the subspace of $C^{\infty }(B̅L,E)$ consisting of all harmonic functions in B, where B is the ball in the n-dimensional Euclidean space and E is any Banach space. Consider also the space $h^{-\infty }(E*)$ consisting of all harmonic E*-valued functions g such that ${(1-|x\left|\right)}^{m}f$ is bounded for some m>0. Then the dual $h^{\infty }(E*)$ is represented by $h^{-\infty }(E*)$ through $⟨f,g{⟩}_0=li{m}_{r\to 1}{ʃ}_B⟨f\left(rx\right),g\left(x\right)⟩dx$, $f\in h^{-\infty }(E*),g\in h^{\infty }\left(E\right)$. This extends the results of S. Bell in the scalar case.

Let X be a complex Banach space and let Bloch(X) denote the space of X-valued analytic functions on the unit disc such that $su{p}_{\left|z\right|<1}(1-|z\left|²\right)\left|\right|f^{\text{'}}\left(z\right)\left|\right|<\infty$. A sequence (Tₙ)ₙ of bounded operators between two Banach spaces X and Y is said to be an operator-valued multiplier between Bloch(X) and ℓ₁(Y) if the map ${\sum }_{n=0}^{\infty }xₙzⁿ\to \left(Tₙ\left(xₙ\right)\right)ₙ$ defines a bounded linear operator from Bloch(X) into ℓ₁(Y). It is shown that if X is a Hilbert space then (Tₙ)ₙ is a multiplier from Bloch(X) into ℓ₁(Y) if and only if $su{p}_k{\sum }_{n=2^k}^{2^{k+1}}\left|\right|Tₙ\left|\right|²<\infty$. Several results about Taylor coefficients of vector-valued...

In this note we study some properties concerning certain copies of the classic Banach space $c_0$ in the Banach space $ℒ\left(X,Y\right)$ of all bounded linear operators between a normed space $X$ and a Banach space $Y$ equipped with the norm of the uniform convergence of operators.

Several conditions are given under which l1 embeds as a complemented subspace of a Banach space E if it embeds as a complemented subspace of an Orlicz space of E-valued functions. Previous results in Pisier (1978) and Bombal (1987) are extended in this way.

For a locally compact Hausdorff space K and a Banach space X let C₀(K, X) denote the space of all continuous functions f:K → X which vanish at infinity, equipped with the supremum norm. If X is the scalar field, we denote C₀(K, X) simply by C₀(K). We prove that for locally compact Hausdorff spaces K and L and for a Banach space X containing no copy of c₀, if there is an isomorphic embedding of C₀(K) into C₀(L,X), then either K is finite or |K| ≤ |L|. As a consequence, if there is an isomorphic embedding...

The space of multilinear mappings of nuclear type (s;r1,...,rn) between Banach spaces is considered, some of its properties are described (including the relationship with tensor products) and its topological dual is characterized as a Banach space of absolutely summing mappings.

In this paper there is proved that every Musielak-Orlicz space is reflexive iff it is $P$-convex. This is an essential extension of the results given by Ye Yining, He Miaohong and Ryszard Płuciennik [16].

It is proved that the Köthe-Bochner function space E(X) has property β if and only if X is uniformly convex and E has property β. In particular, property β does not lift from X to E(X) in contrast to the case of Köthe-Bochner sequence spaces.