On Opial's Lemma

Abstract

Opial's Lemma is a fundamental result in the convergence analysis of sequences generated by optimization algorithms in real Hilbert spaces. We introduce the concept of Opial sequences - sequences for which the limit of the distance to each point in a given set exists. We systematically derive properties of Opial sequences, contrasting them with the well-studied Fej\'er monotone sequences, and establish conditions for weak and strong convergence. Key results include characterizations of weak convergence via weak cluster points (reaffirming Opial's Lemma), strong convergence via strong cluster points, and the behavior of projections onto Opial sets in terms of asymptotic centers. Special cases and examples are provided to highlight the subtle differences in convergence behaviour and projection properties compared to the Fej\'er monotone case.