Related papers: Every separable complex Fr\'echet space with a con…

We discuss the continuity of the composition on several spaces of holomorphic mappings on open subsets of a complex Banach space. On the Fr\'{e}chet space of the entire mappings that are bounded on bounded sets the composition turns to be…

Motivated by recent work on strict deformation quantization of the unit disk and the Riemann sphere, we study the Fr\'echet space structure of the set of holomorphic functions on the complement $\Omega:=\{(z,w)\in \hat{\mathbb{C}}^2\, :\,…

A well-known open question is whether every countable collection of Lipschitz functions on a Banach space X with separable dual has a common point of Frechet differentiability. We show that the answer is positive for some…

We study holomorphic functions attaining weighted norms and its connections with the classical theory of norm attaining holomorphic functions. We prove that there are polynomials on $\ell_p$ which attain their weighted but not their…

We show that the set of Lebesgue integrable functions in $[0,1]$ which are nowhere essentially bounded is spaceable, improving a result from [F. J. Garc\'{i}a-Pacheco, M. Mart\'{i}n, and J. B. Seoane-Sep\'ulveda. \textit{Lineability,…

A separable Banach space $X$ is said to be finitely determined if for each separable space $Y$ such that $X$ is finitely representable (f.r.) in $Y$ and $Y$ is f.r. in $X$ then $Y$ is isometric to $X$. We provide a direct proof (without…

We define a neural network in infinite dimensional spaces for which we can show the universal approximation property. Indeed, we derive approximation results for continuous functions from a Fr\'echet space $\X$ into a Banach space $\Y$. The…

We show that norms on certain Banach spaces $X$ can be approximated uniformly, and with arbitrary precision, on bounded subsets of $X$ by $C^{\infty}$ smooth norms and polyhedral norms. In particular, we show that this holds for any…

We prove that every continuous mapping from a separable infinite-dimensional Hilbert space $X$ into $\mathbb{R}^{m}$ can be uniformly approximated by $C^\infty$ smooth mappings {\em with no critical points}. This kind of result can be…

A metric space $\mathbf{X}$ is called densely complete if there exists a dense set $D$ in $\mathbf{X}$ such that every Cauchy sequence of points of $D $ converges in $\mathbf{X}$. One of the main aims of this work is to prove that the…

This paper complements the work done on simultaneous approximation results in classical Banach spaces, by focusing on approximation by finite Blaschke products. We prove the existence of a finite Blaschke product that approximates a…

We investigate an extended version of Hilbert space of analytic functions called Hilbert space of complex-valued harmonic functions. It is found that functions in Hilbert space of complex-valued harmonic functions exhibit many properties…

We introduce a class of functionals on the space of rapidly decreasing sequences $s$, called $\mathcal{F}_s$-functionals, defined as decomposable sums of quadratic and convex terms with quadratic growth. We prove that such functionals…

We prove compactness of the embeddings in Sobolev spaces for fractional super and sub harmonic functions with radial symmetry. The main tool is a pointwise decay for radially symmetric functions belonging to a function space defined by…

We consider quasi-polynomial spaces of differential forms defined as weighted (with a positive weight) spaces of differential forms with polynomial coefficients. We show that the unisolvent set of functionals for such spaces on a simplex in…

Let $(\varphi_t)_{t\geq 0} $ a semigroup of holomorphic self-maps of the unit disk and $C_t f = f \circ \varphi_t $ the semigroup of composition operators which corresponds to $\varphi_t. $ Given a non-separable Banach space of analytic…

We consider fractional Sobolev spaces $H^\theta$, $\theta\in (0,1)$, on 2D domains and $H^1$-conforming discretizations by globally continuous piecewise polynomials on a mesh consisting of shape-regular triangles and quadrilaterals. We…

Partition functions, also known as homomorphism functions, form a rich family of graph invariants that contain combinatorial invariants such as the number of k-colourings or the number of independent sets of a graph and also the partition…

Classical theorems from the early 20th century state that any Haar measurable homomorphism between locally compact groups is continuous. In particular, any Lebesgue-measurable homomorphism $\phi:\mathbb{R} \to \mathbb{R}$ is of the form…

In this article we prove that every isometric copy of C(L) in C(K) is complemented if L is compact Hausdorff of finite height and K is a compact Hausdorff space satisfying the extension property, i.e., every closed subset of K admits an…