8 problems

Let $J$ be an index set, and give the product space … the product topology. Let $\mathbb{C}^{(J)}=\bigoplus_{j\in J}\mathbb{C}$ denote the space of finite sequences, equipped with…

Abundance conjecture. An infinite-dimensional TVS has infinitely many proper dense subspaces if and only if $w(X)\leq \dim(X)$.

Let $X$ be a topological vector space and let $C\subseteq X$ be a convex set. The quasi-relative interior of $C$, denoted $qri C$, and the face relative interior of $C$, denoted…

Let $(l^{1},\tau^{1})$ be the usual $l^{1}$ space. For each $p>1$, let $i_{p}\colon l^{1}\to l^{p}$ be the inclusion map, and let $i_{w}\colon l^{1}\to l^{1}_{w}$ be the canonical…

Bounded Mackey topology conjecture. For $P=L^{1}\left[0,1\right]$ at least, and possibly for every nonreflexive Banach space $P$, the topology $\mathrm{bm}\left(P^{\ast},P\right)$…

Bounded Mackey topology conjecture. The topology $\mathrm{bm}\left(P^{\ast},P\right)$ should be strictly stronger than $\mathrm{m}\left(P^{\ast},P\right)$, or equivalently should n…

Let $X$ be a non-empty, compact, convex subspace of a topological vector space. A continuous self-map is a map $f\colon X\to X$, and a fixed point of $f$ is a point $x\in X$ such t…

Hypercyclicity conjecture. Any separable ${\mathcal F}$-space with infinite-dimensional dual supports a hypercyclic operator.