Summary: We study the exponent sequences \(\alpha\) for which \(\Lambda_{\infty}(\alpha)\) is embeddable into \(\Lambda_ 1(\alpha)\). We show that in this case \(\alpha\) is necessarily weakly stable but weak stability is not sufficient in general. However, we prove the following:
If \(\Lambda_{\infty}(\alpha)\) is isomorphic to a subspace of \(\Lambda_ 1(\alpha)\) spanned by a block basic sequence then \(\alpha\) is stable, the converse of which is well known. We also give equivalent conditions in terms of embeddings of power series spaces for weak stability of an exponent sequence.