Operator Algebras

p. 2 I.1.1.4: The Riesz-Fischer Theorem is often stated this way today, but neither Riesz nor Fischer (who worked independently) phrased it in terms of completeness of the orthogonal system {e}. If [a, b] is a bounded interval in R, in modern language the original statement of the theorem was that L([a, b]) is complete and abstractly isomorphic to l. According to [Jah03, p. 385], the name “Hilbert space” was first used in 1908 by A. Schönflies, apparently to refer to what we today call l. Von Neumann used the same name for Hilbert spaces in the modern sense (complete inner product spaces), which he defined in 1928.