Related papers: A separable Fr\'echet space of almost universal di…

In this paper we prove a general convergence theorem for almost-additive set functions on unimodular, amenable groups. These mappings take their values in some Banach space. By extending the theory of epsilon-quasi tiling techniques, we set…

Let $X$, $Y$ be two real Banach spaces and $\varepsilon>0$. A standard $\varepsilon$-isometry $f:X\rightarrow Y$ is said to be $(\alpha,\gamma)$-stable (with respect to $T:L(f)\equiv\overline{{\rm span}}f(X)\rightarrow X$ for some $\alpha,…

We show that every metric space with bounded geometry uniformly embeds into an explicit reflexive Banach space (a direct sum of l^p spaces). In the case of discrete groups we show the analogue of a-T-menability. That is, we construct a…

Extending a result of Mashreghi and Ransford, we prove that every complex separable infinite dimensional Fr\'echet space with a continuous norm is isomorphic to a space continuously included in a space of holomorphic functions on the unit…

We prove in this article that every Borelian measure, for example, the distribution of a random variable, in separable Banach space has a support which is compact embedded Banach subspace; and prove that if the norm of the random variable…

A problem of Banach asks whether every infinite-dimensional Banach space which is isomorphic to all its infinite-dimensional subspaces must be isomorphic to a separable Hilbert space. In this paper we prove a result of a Ramsey-theoretic…

Let $\mathfrak{A}$ be a Banach algebra, and $\mathcal{X}$ a Banach $\mathfrak{A}$-bimodule. A bounded linear mapping $\mathcal{D}:\mathfrak{A}\rightarrow \mathcal{X}$ is approximately semi-inner derivation if there eixist nets…

Let $X$, $Y$ be two real Banach spaces, and $\eps\geq0$. A map $f:X\rightarrow Y$ is said to be a standard $\eps$-isometry if $|\|f(x)-f(y)\|-\|x-y\||\leq\eps$ for all $x,y\in X$ and with $f(0)=0$. We say that a pair of Banach spaces…

A metric space $\mathrm{M}=(M,\de)$ is {\em indivisible} if for every colouring $\chi: M\to 2$ there exists $i\in 2$ and a copy $\mathrm{N}=(N, \de)$ of $\mathrm{M}$ in $\mathrm{M}$ so that $\chi(x)=i$ for all $x\in N$. The metric space…

In a companion paper (Studia Math., 2023), we proved for every $\lambda\in(1,2]$ the existence of a $(\lambda^+)$-injective renorming of $\ell_\infty$ that is not $\lambda$-injective, thereby establishing a~forgotten theorem of…

Leveraging the infinite dimensional neural network architecture we proposed in arXiv:2109.13512v4 and which can process inputs from Fr\'echet spaces, and using the universal approximation property shown therein, we now largely extend the…

In this paper we explore some basic properties of quasi-Banach function spaces which are important in applications. Namely, we show that they posses a generalised version of Riesz--Fischer property, that embeddings between them are always…

In this paper we present a simple proof of Gowers Dichotomy which states that every infinite dimensional Banach Space has a subspace which either contains an unconditional basic sequence or is hereditarily indecomposable. Our approach is…

Recently, versions of neural networks with infinite-dimensional affine operators inside the computational units (``neural operator'' networks) have been applied to learn solutions to differential equations. To enable practical computations,…

This work performs a study of the category of complete matrix-normed spaces, called matricial Banach spaces. Many of the usual constructions of Banach spaces extend in a natural way to matricial Banach spaces, including products, direct…

We introduce the categories of quasi-measurable spaces, which are slight generalizations of the category of quasi-Borel spaces, where we now allow for general sample spaces and less restrictive random variables, spaces and maps. We show…

We show that there exist infinite-dimensional extremely non-complex Banach spaces, i.e. spaces $X$ such that the norm equality $\|Id + T^2\|=1 + \|T^2\|$ holds for every bounded linear operator $T:X\longrightarrow X$. This answers in the…

We construct infinitely differentiable norms and partitions of unity for a class of Banach spaces which includes all spaces $\C(K)$ with $K$ a countable compact space, and all spaces $\C_0[0,\Omega )$ with $\Omega $ an ordinal.

We carry out a systematic study of decidability for theories of (a) real vector spaces, inner product spaces, and Hilbert spaces and (b) normed spaces, Banach spaces and metric spaces, all formalised using a 2-sorted first-order language.…

Let $S$ be a closed subset of a Banach space $X$. Assuming that $S$ is epi-Lipschitzian at $\bar{x}$ in the boundary $ \bd S$ of $S$, we show that $S$ is strictly Hadamard differentiable at $\bar{x}$ IFF the Clarke tangent cone $T(S,…