Weak sequential completeness of sequence spaces

In the [4] Köthe and Toeplitz introduced the theory of sequence spaces and established many of the basic properties of sequence spaces by using methods of classical analysis. Later many of these same properties sequence spaces were reestablished by using “soft proofs” of functional analysis. In this note we would like to point out that an improved version of a classical lemma of Schur due to Hahn can be used to give very short proofs of two of the weak sequential completeness results of Köthe and Toeplitz. One of our proofs actually gives an improvement of one of the completeness results of Köthe and Toeplitz which was obtained by Benett usinf functional analysis methods and the method or proof is used in $\S3$ to obtain a completeness result for $\beta$-duals of vector-valued sequence spaces. One of our completeness results is employed to obtain a more general form of a Hellinger-Toeplitz type theorem for sequence spaces due to Köthe and the second completeness result is employed to obtain another Hellinger-Toeplitz type theorem for sequence spaces which covers additional cases not covered by Köthe’s result.