Related papers: When is a locally convex space Eberlein-Grothendie…

Smooth functions $f:G\to E$ from a topological group $G$ to a locally convex space $E$ were considered by Riss (1953), Boseck, Czichowski and Rudolph (1981), Belti\c{t}\u{a} and Nicolae (2015), and others, in varying degrees of generality.…

In our paper [18] we showed that a Tychonoff space $X$ is a $\Delta$-space (in the sense of [20], [30]) if and only if the locally convex space $C_{p}(X)$ is distinguished. Continuing this research, we investigate whether the class $\Delta$…

We characterize Ascoli spaces by showing that a Tychonoff space $X$ is Ascoli iff the canonical map from the free locally convex space $L(X)$ over $X$ into $C_k\big(C_k(X)\big)$ is an embedding of locally convex spaces. We prove that an…

Let $Y$ be a metrizable space containing at least two points, and let $X$ be a $Y_{\mathcal{I}}$-Tychonoff space for some ideal $\mathcal{I}$ of compact sets of $X$. Denote by $C_{\mathcal{I}}(X,Y)$ the space of continuous functions from…

We prove local Poincar\'e inequalities under various curvature-dimension conditions which are stable under the measured Gromov-Hausdorff convergence. The first class of spaces we consider is that of weak CD(K,N) spaces as defined by Lott…

A topological space $G$ is said to be a {\it rectifiable space} provided that there are a surjective homeomorphism $\varphi :G\times G\rightarrow G\times G$ and an element $e\in G$ such that $\pi_{1}\circ \varphi =\pi_{1}$ and for every…

There is a hierarchy of structure conditions for convex sets. In this paper we study a recently defined [3, 8, 9] condition called locally nonconical convexity (abbreviated LNC). Is is easy to show that every strictly convex set is LNC, as…

It has been proved in [J.-D. Hardtke, J. Math. Phys. Anal. Geom. 16, no.2, 119--137 (2020)] that a K\"othe-Bochner space $E(X)$ is locally octahedral/locally almost square if $X$ has the respective property and the simple functions are…

We start the systematic study of Fr\'{e}chet spaces which are $\aleph$-spaces in the weak topology. A topological space $X$ is an $\aleph_0$-space or an $\aleph$-space if $X$ has a countable $k$-network or a $\sigma$-locally finite…

In many applications it is important to establish if a given topological preordered space has a topology and a preorder which can be recovered from the set of continuous isotone functions. Under antisymmetry this property, also known as…

A topological space is locally equiconnected if there exists a neighborhood $U$ of the diagonal in $X\times X$ and a continuous map $\lambda:U\times[0,1]\to X$ such that $\lambda(x,y,0)=x$, $\lambda(x,y,1)=y$ et $\lambda(x,x,t)=x$ for…

Let $E$ be a Banach space such that $E'$ has the Radon-Nikod\'ym property. The aim of this work is to connect relative weak compactness in the $E$-valued martingale Hardy space $H^{1}(\mu,E)$ to a convex compactness criterion in a weaker…

A topological space $X$ has the strong Pytkeev property at a point $x\in X$ if there exists a countable family $\mathcal N$ of subsets of $X$ such that for each neighborhood $O_x\subset X$ and subset $A\subset X$ accumulating at $x$, there…

There are presented certain results on extending continuous linear operators defined on spaces of E-valued continuous functions (defined on a compact Hausdorff space X) to linear operators defined on spaces of E-valued measurable functions…

For an extended locally convex space (elcs) $(X,\tau)$, the authors in [10] studied the topology $\tau_{ucb}$ of uniform convergence on bounded subsets of $(X,\tau)$ on the dual $X^*$ of $(X,\tau)$. In the present paper, we use the topology…

Our main result states that the hyperspace of convex compact subsets of a compact convex subset $X$ in a locally convex space is an absolute retract if and only if $X$ is an absolute retract of weight $\le\omega_1$. It is also proved that…

A Banach space $E$ has the Grothendieck property if every (linear bounded) operator from $E$ into $c_0$ is weakly compact. It is proved that, for an integer $k>1$, every $k$-homogeneous polynomial from $E$ into $c_0$ is weakly compact if…

This paper introduces first order Sobolev spaces on certain rectifiable varifolds. These complete locally convex spaces are contained in the generally nonlinear class of generalised weakly differentiable functions and share key functional…

Motivated by ill-posed PDEs such as $\mathrm{div} (v) = F$ we study locally convex topologies $\mathcal{T}_{\mathcal{C}}$ on real vector spaces $X$ that are a ``localized'' version of a locally convex topology $\mathcal{T}$ to members of a…

For a Tychonoff space $X$, let $C_k(X)$ and $C_p(X)$ be the spaces of real-valued continuous functions $C(X)$ on $X$ endowed with the compact-open topology and the pointwise topology, respectively. If $X$ is compact, the classic result of…