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.
The main purpose of the present paper is to extend the theory of non-smooth atomic decompositions to anisotropic function spaces of Besov and Triebel-Lizorkin type. Moreover, the detailed analysis of the anisotropic homogeneity property is carried out. We also present some results on pointwise multipliers in special anisotropic function spaces.
We give a full characterization of normability of Lorentz spaces . This result is in fact known since it can be derived from Kamińska A., Maligranda L., On Lorentz spaces, Israel J. Funct. Anal. 140 (2004), 285–318. In this paper we present an alternative and more direct proof.
We study normability properties of classical Lorentz spaces. Given a certain general lattice-like structure, we first prove a general sufficient condition for its associate space to be a Banach function space. We use this result to develop an alternative approach to Sawyer’s characterization of normability of a classical Lorentz space of type . Furthermore, we also use this method in the weak case and characterize normability of . Finally, we characterize the linearity of the space by a simple...
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.
Every Orlicz space equipped with Orlicz norm has weak sum property, therefore, it has weakly normal structure and fixed point property. A criterion of sum property also of normal structure for such spaces is given as well, which shows that every Orlicz space has isonormal structure.
Let ϕ: ℝ → ℝ₊ ∪ 0 be an even convex continuous function with ϕ(0) = 0 and ϕ(u) > 0 for all u > 0 and let w be a weight function. u₀ and v₀ are defined by u₀ = supu: ϕ is linear on (0,u), v₀=supv: w is constant on (0,v) (where sup∅ = 0). We prove the following theorem. Theorem. Suppose that (respectively, ) is an order continuous Lorentz-Orlicz space. (1) has normal structure if and only if u₀ = 0 (respectively, (2) has weakly normal structure if and only if .