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If one studies the paper written by Banach in which he first proved his fixed point theorem one would find that he did not prove the theorem for abstract complete metric spaces. He proved it for the special case of complete normed vector spaces.

My question is - who extended his result to complete metric spaces?

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Pages 97-107 of the book Mathematical Lives: Protagonists of the Twentieth Century From Hilbert to Wiles is an article by Guerraggio about Caccioppoli. On p. 100, Guerraggio says the contraction mapping theorem for complete metric spaces is due to Caccioppoli, who is said to have been almost alone in publishing in Banach-style functional analysis in the 1930s. Specifically, Guerraggio says that in Caccioppoli's short paper "Un teorema generale sull’esistenza di elementi uniti in una transformazione funzionale", Rend. Accad. Naz. Lincei, 11 (1930), 794–799 the fixed point theorem for contractions on complete metric spaces appears and all that is written there for a proof is the statement that the proof is obvious. Locate a copy of this paper to see if it answers your question.

On p. 176 of Italian Mathematics Between the Two World Wars by Guerraggio and Nastasi, Caccioppoli's formulation of the fixed point theorem on complete metric spaces appears in Italian near the bottom of the page and a translation into English is in a footnote on that page.

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