On absolute almost convergence

Lack of Tauberian theorem for absolute almost convergence

“…Though b is not closed in ∞ , b is not b c contrary to our expectation in [7]. For other topological properties, see [9].…”

“…P r o o f . There exists P a n ∈ b such that P a n / ∈ c. See [8], where an example P a n ∈ b is given with a n → 0 and so P a n / ∈ c. Again, the series…”

“…At this stage we note that the convergence of P |a n |/n is necessary for P a n to be in b (see [7]). …”

“…|φ m, n (a)| converges uniformly in n. Note that it was proved in [7] that the convergence of P m |ϕ m,n | for only one n implies convergence for any other values of n. We denote the set of all absolutely convergent sequences and absolutely almost convergent sequences respectively by and b . It is known (see [6]) that b is a Banach space with the norm defined by…”

“…where {n r } is an increasing sequence of positive integers (tending to ∞ as r → ∞) to be chosen as follows : Since χ(n) ↓ 0, we can find an increasing sequence {n r } of positive integers (see [7]) such that…”