We show that if U is a domain of existence in a separable Banach space, then the set of holomorphic functions on U whose domain of existence is U is lineable and algebrable.

Let U be an open subset of a separable Banach space. Let ℱ be the collection of all holomorphic mappings f from the open unit disc 𝔻 ⊂ ℂ into U such that f(𝔻) is dense in U. We prove the lineability and density of ℱ in appropriate spaces for different choices of U.

It is shown that if E is a Frechet space with the strong dual E* then Hb(E*), the space of holomorphic functions on E* which are bounded on every bounded set in E*, has the property (DN) when E ∈ (DN) and that Hb(E*) ∈ (Ω) when E ∈ (Ω) and either E* has an absolute basis or E is a Hilbert-Frechet-Montel space. Moreover the complementness of ideals J(V) consisting of holomorphic functions on E* which are equal to 0 on V in H(E*) for every nuclear Frechet space E with E ∈ (DN) ∩ (Ω) is stablished...

Estudiamos algunas situaciones donde encontramos un problema que, a primera vista, parece no tener solución. Pero, de hecho, existe un subespacio vectorial grande de soluciones del mismo.

For U open in a locally convex space E it is shown in [31] that there is a complete locally convex space G(U) such that $G{\left(U\right)}_i^{\text{'}}=(\mathscr{H}\left(U\right),{\tau }_{\delta })$. Here, we assume U is balanced open in a Fréchet space and give necessary and sufficient conditions for G(U) to be Montel and reflexive. These results give an insight into the relationship between the ${\tau }_0$ and ${\tau }_{\omega }$ topologies on ℋ (U).

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.

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...

Let $D^{\text{'}}\subset {ℂ}^{n-1}$ be a bounded domain of Lyapunov and $f(z^{\text{'}},z_n)$ a holomorphic function in the cylinder $D=D^{\text{'}}\times U_n$ and continuous on $\overline{D}$. If for each fixed $a^{\text{'}}$ in some set $E\subset \partial D^{\text{'}}$, with positive Lebesgue measure $\text{mes}E>0$, the function $f(a^{\text{'}},z_n)$ of $z_n$ can be continued to a function holomorphic on the whole plane with the exception of some finite number (polar set) of singularities, then $f(z^{\text{'}},z_n)$ can be holomorphically continued to $(D^{\text{'}}\times ℂ)\setminus S$, where $S$ is some analytic (closed pluripolar) subset of $D^{\text{'}}\times ℂ$.

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.

Si studiano «combinazioni convesse complesse» per mappe olomorfe dal disco unità di $ℂ$ in un dominio convesso limitato $D$ di uno spazio di Banach complesso $E$, e se ne traggono conseguenze sul carattere globale della non unicità per le geodetiche complesse di $D$.