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Let 1 ≤ p < ∞ and 1 ≤ r ≤ p * , where p * is the conjugate index of p. We prove an omnibus theorem, which provides numerous equivalences for a sequence (x n ) in a Banach space X to be a ( p, r )-null sequence. One of them is that (x n ) is ( p, r )-null if and only if (x n ) is null and relatively ( p, r )-compact. This equivalence is known in the "limit" case when r = p * , the case of the p-null sequence and p-compactness. Our approach is more direct and easier than those applied for the proof of the latter result. We apply it also to characterize the unconditional and weak versions of ( p, r )-null sequences.

We introduce the notion of (p; r)-null sequences in a Banach space and we prove a Grothendieck-like result: a subset of a Banach space is relatively (p; r)-compact if and only if it is contained in the closed convex hull of a (p; r)-null sequence. This extends a recent description of relatively p-compact sets due to Delgado and Piñeiro, providing to it an alternative straightforward proof.

It is proved that a non-archimedean FK-space E includes the space c0 of null-sequences if and only if E includes the space l1` of absolutely summable sequences, and for both inclusions the boundedness in E of the set {ek | k ∈ N} of the unit vectors is necessary and sufficient.It is shown that in the non-archimedean context the concepts of wedge space and semiconservative space are equivalent.

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