A complete space, in the metric sense, is a complete metric space: every Cauchy sequence has a limit in the space. Completeness refers to the specified metric (Axler 2026, p. 151). A Banach space is complete for the metric induced by its norm, and a Hilbert space is complete for the norm induced by its inner product.
Completeness of a complete orthogonal system has a different meaning: its vector space span is dense in the ambient Hilbert space. It is a property of the system within the space, rather than the assertion that its members form a complete metric space (Axler 2026, pp. 242-244).
