On generalized sequence spaces defined by modulus functions

Abstract

Abstract. Let X,| ·| be a seminormed space, Φ = (φ k ) a sequence of moduli, and B a sequence of infinite scalar matrices B i = (b i kj ). Let (λ, g λ ) and (Λ, g Λ ) be solid F-seminormed (paranormed) spaces of single and double number sequences, respectively. V. Soomer and E. Kolk proved in 1996-1997 that the set of all scalar sequences u = (u k ) with Φ(u) = (φ k (|u k |)) ∈ λ is a linear space which may be topologized by the F-seminorm (paranorm) g λ,Φ (u) = g λ (Φ(u)) under certain restrictions on Φ or (λ, g λ ). We generalize this result to the space of all X-valuedApplications are given in the case when Λ is the strong summability domain of a non-negative matrix method. Our corollaries and critical remarks outline results from more than thirty previous papers by many different authors.

On Generalized Orlicz Sequence Spaces Defined by Double Sequences

“…An F-seminormed DS space (Λ, g) is called a double AK-space (see, for example, [13]), if Λ contains the sequences e (2) k and for any…”

B p F q s g Sequence Space On The Spaces With Seminorm

“…Connor (1989), Maddox (1986)'nın tanımını Cesàro matrisi yerine herhangi negatif olmayan regüler matris alarak   , w A f toplanabilme metoduna genelleştirdi. Şahiner (2002) Şunu belirtelim ki, modulus fonksiyon yardımıyla bir çok uzay oluşturulmuştur (Banerji and Galiz, 2000;Soomer, 2000;Esi, 2000;Kolk, 1997Kolk, , 1990Kolk, , 1998Kolk, , 1999Kolk, , 2013Esi and Et, 1996;Pehlivan and Fisher, 1995;Gupta and Bhola, 1975;Bilgin, 1994Bilgin, , 1996Bilgin, , 2004Bilgin ve Altun, 2007;Raj and Sharma, 2011;Karakaya ve Şimşek, 2004;Bhardwaj and Bala, 2009; Işık, 2011 ve diğerleri).…”