On the Euler sequence spaces which include the spaces ℓp and ℓ∞ I

In the present paper, we introduce the Euler sequence space epr consisting of all sequences whose Euler transforms of order r are in the space ℓp of non-absolute type which is the BK-space including the space ℓp and prove that the spaces epr and ℓp are linearly isomorphic for 1 ⩽ p ⩽ ∞. Furthermore, we give some inclusion relations concerning the space epr. Finally, we determine the α-, β- and γ  -duals of the space epr for 1 ⩽ p ⩽ ∞ and construct the basis for the space epr, where 1 ⩽ p < ∞.