It is shown that if (S,∑,m) is an atomless finite measure space and X is a Banach space without the Radon-Nikodym property, then the quotient space cabv(∑,m;X)/L¹(m;X) is nonseparable.

For each natural number N, we give an example of a Banach space X such that the set of norm attaining N-linear forms is dense in the space of all continuous N-linear forms on X, but there are continuous (N+1)-linear forms on X which cannot be approximated by norm attaining (N+1)-linear forms. Actually,X is the canonical predual of a suitable Lorentz sequence space. We also get the analogous result for homogeneous polynomials.

We give a polynomial version of Shmul'yan's Test, characterizing the polynomials that strongly attain their norm as those at which the norm is Fréchet differentiable. We also characterize the Gateaux differentiability of the norm. Finally we study those properties for some classical Banach spaces.

Let $f$ be a continuous map of the closure $\overline{\mathrm{\Delta }}$ of the open unit disc $\mathrm{\Delta }$ of $\mathbb{C}$ into a unital associative Banach algebra $\mathcal{A}$, whose restriction to $\mathrm{\Delta }$ is holomorphic, and which satisfies the condition whereby $0\notin \sigma \left(f\left(z\right)\right)\subset \overline{\mathrm{\Delta }}$ for all $z\in \mathrm{\Delta }$ and $\sigma \left(f\left(z\right)\right)\subset \partial \mathrm{\Delta }$ whenever $z\in \partial \mathrm{\Delta }$ (where $\sigma \left(x\right)$ is the spectrum of any $x\in \mathcal{A}$). One of the basic results of the present paper is that $f$ is , that is to say, $\sigma \left(f\left(z\right)\right)$ is then a compact subset of $\partial \mathrm{\Delta }$ that does not depend on $z$ for all $z\in \overline{\mathrm{\Delta }}$. This fact will be applied to holomorphic self-maps of the open unit ball of some $J^{*}$-algebra...

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.

An example of a nonzero σ-finite Borel measure μ with everywhere dense linear manifold ${}_{\mu }$ of admissible (in the sense of invariance) translation vectors is constructed in the Hilbert space ℓ₂ such that μ and any shift ${\mu }^{\left(a\right)}$ of μ by a vector $a\in \ell ₂{\setminus }_{\mu }$ are neither equivalent nor orthogonal. This extends a result established in [7].

A class of convex functions where the sets of subdifferentials behave like the unit ball of the dual space of an Asplund space is found. These functions, which we called Asplund functions also possess some stability properties. We also give a sufficient condition for a function to be an Asplund function in terms of the upper-semicontinuity of the subdifferential map.

Let $𝒜$ be a $\sigma$-algebra on a set $X$. If $A$ belongs to $𝒜$ let $e\left(A\right)$ be the characteristic function of $A$. Let ${\ell }_0^{\infty }(X,𝒜$ be the linear space generated by $\left\{e\right(A):A\in 𝒜\}$ endowed with the topology of the uniform convergence. It is proved in this paper that if $\left(E_n\right)$ is an increasing sequence of subspaces of ${\ell }_0^{\infty }(X,𝒜)$ covering it, there is a positive integer $p$ such that $E_p$ is a dense barrelled subspace of ${\ell }_0^{\infty }(X,𝒜)$, and some new results in measure theory are deduced from this fact.