107 problems

Let $X$ be the Banach space underlying a linear discrete-time system, and let $A_m^k$ denote its associated evolution operator from time $k$ to time $m$. The system is uniformly ex…

Is every Banach space $X$ with the renorming-stable ball-covering property separable? More precisely, if for every norm $\lVert\cdot\rVert'$ equivalent to the given norm on $X$, th…

For every integer $N\ge 1$ and every linear subspace $V\subset \ell_\infty^N$ with absolute projection constant $\lambda(V)<\frac{4}{3}$, one has …

Let $E$ and $F$ be separable symmetric Banach sequence spaces, and let $C_E$ and $C_F$ denote the corresponding ideals of compact operators on a separable Hilbert space. Does…

For every pointed metric space $(M,p)$, the Lipschitz-free space $\mathcal{F}(M)$ is the strongly unique isometric predual of $\operatorname{Lip}_0(M)$; that is,…

Let $X$ be a Banach space over $\mathbb{R}$ or $\mathbb{C}$ having an unconditional Schauder basis, and let $Y$ be a complemented closed subspace of $X$, meaning that there exists…

Determine whether, for every complete pointed metric space $(M,d,0)$, the Banach space $\operatorname{Lip}_0(M)$ of real-valued Lipschitz functions vanishing at $0$ has a strongly…

Does there exist an infinite-dimensional Banach space $X$ such that every bounded linear operator $T\in\mathcal{L}(X,X)$ attains its norm; that is, for every $T\in\mathcal{L}(X,X)$…

Let $V$ be a normed space over the reals, and let $2\leq k<\dim V$ be an integer. Suppose that all linear $k$-dimensional subspaces of $V$ are isometric to each other. Banach's iso…

Let $G$ be a Gromov hyperbolic group, and consider affine actions of $G$ on a Hilbert space whose linear parts are uniformly bounded representations. An affine action is proper whe…

Let $({\mathcal M},\rho)$ be a pseudometric space, let $X$ be a Banach space, and let $m$ be a fixed positive integer. Set … For a set-valued mapping…

Let $X$ and $Y$ be Banach-space-valued random variables, and let $f$ be a similarity transformation of the Banach space. The Banach-space perfect-correlation conjecture. The axiom…

Let $X$ be a separable Banach space that is not isomorphic to a Hilbert space. Recall that $X$ is ergodic if the equivalence relation $\mathbf{E}_0$ is Borel-reducible to the isomo…

Let $A$ be the generator of a $C_0$-group on $\ell_p$, with $p>1$, whose eigenvalues, counted with multiplicity, are $\{\lambda_n\}_{n=1}^{\infty}$ and whose corresponding normaliz…

Let $X$ be a $K$-convex Banach space. The property $\mathrm{GL}_2$ is the Gordon–Lewis property with exponent $2$, and a Banach lattice has finite cotype in the usual Banach-space…

Let $X$ be a separable Banach space, let $A\subseteq X$ be Borel, and let $\mathbb{A}_{\mathsf{G}}$ be the auxiliary game referred to in the paper, with $\mathbb{A}$ denoting the c…

Let $X$ be a separable Banach space over $cmathbb{R}$, let $A\toolbox X$ be Borel and Aronszajn-null, and let $\mathcal{A}_{\mathsf{G}}$ be the game in which player $\mathbf{I}$ ch…

Let $X$ be a Banach space with a basis, respectively an unconditional basis, which is not equivalent to the canonical basis of $c_0$ or $\ell_p$ for $1\leq p<+\infty$. A normalized…

Let $X$ be a Banach space with an unconditional basis. A block-subspace is a closed subspace generated by a block-sequence of that basis, and $E_0$ is the equivalence relation of e…

A separable Banach space is a Banach space with a countable dense subset, and a Banach space is ergodic when the isomorphism relation between its infinite-dimensional subspaces is…

Let $G=\prod_{i=1}^{m}\mathbf{G}_{i}(k_{i})$, where each $k_i$ is a local field and each $\mathbf{G}_{i}(k_i)$ is the group of $k_i$-points of a Zariski connected simple $k_i$-alge…

Three-type existence claim. Every infinite-dimensional Banach space $\mathfrak X$ admits a bounded biorthogonal system $\left\{x_n,x_n^*\right\}$ that is one of the following three…

Let $X$ be a finite-dimensional Banach space, with $n=\dim X$. Finite-dimensional characterization conjecture. (a) If every one-dimensional subspace of $X$ can be represented as an…

Let $X$ be a strictly monotone Banach space with a $1$-unconditional basis, sufficiently different from a Hilbert space. Block-structure conjecture. Every norm-one projection on…

Let $X$ be a rearrangement-invariant function space on $[0,1]$, so that $X$ contains the constant functions, and let $P$ be a contractive projection on $X$ satisfying…