Abel summability method is a regular method of sequential convergence in this manner. Any matrix summability method on a subspace of s is a method of sequential convergence. A function f is called G-continuous (see also ) if G( f( p)) = f( G( p)) for any G-convergent sequence p.
A method G is called subsequential if whenever p is G-convergent with G( p) = l, then there is a subsequence ( p n k) of p with lim k p n k = l. A method G is called regular if every convergent sequence p = ( p n) is G-convergent with G( p) = lim p. A sequence p = ( p n) is said to be G-convergent to l if p ∈ c G, and G( p) = l. The concept of continuity and any concept involving continuity play a very important role not only in pure mathematics but also in other branches of sciences involving mathematics especially in computer sciences, information theory, and dynamical systems.Ī method of sequential convergence is a linear function G defined on a linear subspace of s, denoted by c G, into R where R and s denote the set of real numbers and the space of all sequences, respectively.
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