6 problems

Conditional-basis conjecture. For nonreal coupling constant, the eigenfunctions of the harmonic oscillator do not form a conditional basis.

Let $1<p<\infty$, and let $U=\{u_k\}$ be an exponential-function family in $L^p(0,1)$, with the distance between normalized exponential-function families defined by the displayed q…

Let $(e_n)_{n\in\mathbb N}$ be a Schauder basis. It is a Rosenthal basis if every normalized block-sequence of $(e_n)_{n\in\mathbb N}$ has a subsequence equivalent to…

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…

Quasi-equivalence conjecture. All bases in a nuclear Fréchet space should be quasi-equivalent.