Frigyes Riesz (1880-1956)

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Riesz, F., Sur certains systemes d'equations fonctionelles et l'approximation des fonctions continues, Comptes Rendus Acad. Sci. Paris 150 (1910), 647-677. Also appears in Oeuvres of F. Riesz on p. 403-406. There is a mix-up in some copies of the Oeuvres and there this paper is on pages 403, 404, 398 and 399. This is the first paper where it is stated and proved that an element of C([a,b]) is in the closure of a subspace if and only if every continuous linear functional that vanishes on the subspace also vanishes on the element.

Riesz, F., Über lineare Funktionalgleichungen, Acta Math. 41 (1918), 71-98. Contains first general proof of existence of best approximation from finite-dimensional subspace (see Hilfssatz 3, p. 77, i.e., Proposition 3 in the English translation of the relevant part of the paper).