A necessary condition for Kostyuchenko type systems and system of powers to be a basis in (1 ≤ p < +∞) spaces is obtained. In particular, we find a necessary condition for a Kostyuchenko system to be a basis in (1 ≤ p < +∞).
We show that an infinite-dimensional complete linear space X has: ∙ a dense hereditarily Baire Hamel basis if |X| ≤ ⁺; ∙ a dense non-meager Hamel basis if for some cardinal κ.
We show that in for p ≠ 2 the constants of equivalence between finite initial segments of the Walsh and trigonometric systems have power type growth. We also show that the Riemann ideal norms connected with those systems have power type growth.
We prove that if 𝓒 is a family of separable Banach spaces which is analytic with respect to the Effros Borel structure and no X ∈ 𝓒 is isometrically universal for all separable Banach spaces, then there exists a separable Banach space with a monotone Schauder basis which is isometrically universal for 𝓒 but not for all separable Banach spaces. We also establish an analogous result for the class of strictly convex spaces.
Let K be a non-archimedean valued field which contains Qp and suppose that K is complete for the valuation |·|, which extends the p-adic valuation. Vq is the closure of the set {aqn|n = 0,1,2,...} where a and q are two units of Zp, q not a root of unity. C(Vq → K) is the Banach space of continuous functions from Vq to K, equipped with the supremum norm. Our aim is to find normal bases (rn(x)) for C(Vq → K), where rn(x) does not have to be a polynomial.
Relations between different notions measuring proximity to ℓ₁ and distortability of a Banach space are studied. The main result states that a Banach space all of whose subspaces have Bourgain ℓ₁-index greater than , α < ω₁, contains either an arbitrarily distortable subspace or an -asymptotic subspace.