Topologies in generalized Orlicz sequence spaces; Filomat

In 1994 S.D. Parashar and B. Choudhary defined certain paranorms in some Orlicz sequence spaces of Maddox type. Their ideas are applied later by many authors for topologization of various generalized Orlicz sequence spaces. We determine alternative F-seminorms in such spaces by using the standard arguments of modular spaces theory and a result about the topologization of sequence spaces defined by modulus functions. 1 Notation and background Let N = {1, 2,...} and let K be the field of real numbers R or complex numbers C. We write limn, supn, infn and n instead of limn→∞, supn∈N, infn∈N and∑∞ n=1, respectively. By the symbol ι we denote the identity mapping ι(z) = z. The superposition of two mappings f and g is denoted by fg, i.e., (fg)(z) = f(g(z)). For two sequences x = (xk), y = (yk) we use the notation xy = (xkyk) provided that xkyk is determined for all k ∈ N. We also use the notation R+ = [0,∞) and ek = (eik)i∈N (k ∈ N), where eik = 1 if i = k and eik = 0 otherwise. In all definitions which contain infinite series we tacitly assume the convergence of these series. An F-space is usually understood as a complete metrizable topological vector space over K. It is known that the topology of an F-space E can be given by an F-norm, i.e., by the functional g: E → R with the axioms (see [20], p. 13) (N1) g(0) = 0; (N2) g(x+ y) ≤ g(x) + g(y) (x, y ∈ E); (N3) |α | ≤ 1 (α ∈ K), x ∈ E = ⇒ g(αx) ≤ g(x)