On Linear Functionals and Summability Factors for Strong Summability II

Extract

The first part of this paper, which will be referred to by I, appeared in Volume 30 of this journal. The present paper will use the same bibliography as I.

Theorem 1 in I shows that the knowledge of all continuous linear functionals in o[A]_p is essential in determining convergence and summability factors for strong summability. So far, Theorem 7 in I was for a general A the only tool in deciding whether a given sequence ∈ generates such a functional. We mentioned in a remark following Theorem 7 the difficulties in verifying the conditions of this theorem (two parameters are involved). In the present paper, we study continuous linear functionals in o[A]₁ in more detail, and we obtain in a corollary to Theorem 22 a condition which appears to be a more satisfactory answer to the question, whether a given sequence ∈ generates a continuous linear functional in o[A]₁.