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By Hestenes M.R.

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5) with M(f'(a), ~1' x) (:~~:~) =0 , ~ 1- o. x The composition of two M. Then x J M(f,~, t)'lj;(t)dt (M,;, 0 M

1V~m-~(a, 1) I" ~ 0(1), n ~ v:(",-1) ( k k a, 1)} is slowly oscillating. 12) 1. 12 Let {Sn(an be (A,m)-summable. If for some integer m;::: 1 v~O) (v(m-l)(a, 1), 1) the series {Sn (a = 0(1), n ~ 00 n is (C, m) -summable to its (A, m) -sum. 13) Proof. 13) it follows that f3 O"~m)(S(a)) = L : -L n n k ~ 1 f3k+l k=l k=l for some bounded sequence {f3n}. Since {f3n} is bounded, the last identity implies that {O"~m)(S(a)n is slowly oscillating. • For the generalized Abel's summability method (A, m) we have the generalized Littlewood theorem.

On the other hand, if (Xi, Ii) is a complete biorthogonal sequence for any topological vector space X then the space S (Xi, Ii) consisting of all sequences Sx = (fi (x))i is (algebraically) isomorphic to X under the correspondence T (x) = (Ii (x)) because each Ii is linear and each Ii (x) is zero only for x = O. If we define a topology on S (Xi, Ii) by letting U be open in S (Xi, Ii) if and only if T-l (U) is open in X, then the operator T becomes a topological isomorphism, and S (Xi, Ii) becomes a K space that has AD.

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