Generalized Difference Spaces of Non‐Absolute Type of Convergent and Null Sequences

Abstract

The aim of the present paper is to introduce the spaces and of generalized difference sequences which generalize the paper due to Mursaleen and Noman (2010). These spaces are theBK-spaces of non-absolute type and norm isomorphic to the spaces and , respectively. Furthermore, we derive some inclusion relations determine the , , and duals of those spaces, and construct their Schauder bases. Finally, we characterize some matrix classes from the spaces , and to the spaces , , andc.

Some New Generalized Difference Spaces of Nonabsolute Type Derived from the Spaces ℓp and ℓ∞

“…The inclusion c 0 λ ( B ) ⊂ c λ ( B ) strictly holds by Theorem 3.1 of [20]. So, it is enough to show that the inclusions ℓ p λ ( B ) ⊂ c 0 λ ( B ) and c λ ( B ) ⊂ ℓ ∞ λ ( B ) are strict, where 0 < p < ∞ .…”

Some new paranormed sequence spaces and core theorems

“…letting n → ∞ in (24) and using (22), (23) we have ∑ ∞ k=0 c nk x k → p, as n → ∞. This implies that C ∈ (st ∩ ∞ : c λ ) reg .…”

“…, (see [24]). (iii) If p = e and r = 1, s = −1; then c 0 (λ , Δ; p) = c λ 0 (Δ) and c(λ , B; p) = c λ (Δ), (see [25]).…”

Survey on the domain of the matrix lambda in the normed and paranormed sequence spaces

“…It is immediate by (2:3) that the sets c 0 (B) and c (B) are linear spaces with coordinatewise addition and scalar multiplication, that is c 0 (B) and c (B) are the spaces of generalized di¤erence sequences. Sönmez and Başar [46] have proved that these spaces are the BK spaces of non-absolute type and norm isomorphic to the spaces c 0 and c, respectively. …”