It is well known that Banach's fixed-point theorem was initially conceived as a fixed-point theorem for applications defined in normed spaces (see [1]).

This theorem was conceived in 1922 by Stefan Banach (1892 - 1945) in a famous effort to unify several techniques that forced the convergence of recursive sequences whose usefulness was to prove the existence of solutions for differential equations.

According to [2] it was Felix Hausdorff (1868 - 1942) who in 1914 conceived the notion of metric space in his publication Grundzüge der Mengenlehreem (see [3]).

Question: Who extended the Banach fixed point theorem from the context of normed spaces to the context of metric spaces? What is the original reference that proves the answer?

Note. Some authors say that a primary early example of an extension of Banach's principle is a theorem of Caccioppoli [4] which asserts that the Picard iterates of a mapping $T$ converge in a complete metric space $M$ provided for each $n \geq 1,$ there exists a constant $c_{n}$ such that $$ d\left(T^{n}(x), T^{n}(y)\right) \leq c_{n} d(x, y) $$ for all $x, y \in M,$ where $\sum_{n=1}^{\infty} c_{n}<\infty$.

But, unless I am mistaken, the access I had to the publication [4] does not allow to conclude this statement. Although it is in Italian, the reference seems to be limited to differential equations in normed spaces.

[1] Banach, Stefan. "Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales." Fund. math 3.1 (1922): 133-181.

[2] https://mathshistory.st-andrews.ac.uk/Biographies/Hausdorff/

[3] Hausdorff, Felix. Grundzüge der mengenlehre. Vol. 7. von Veit, 1914.

[4] R. Caccioppoli, Una teorema generale sull'esistenza di elementi uniti in una transformazione funzionale, Ren. Accad. Naz Lincei 11(1930), 794-799 .

  • $\begingroup$ Since the proof for the metric space is the same as Banach's original proof, what is the point of assigning any separate credit for it? $\endgroup$ Feb 10, 2021 at 22:49
  • 2
    $\begingroup$ Hausdorff did not create metric spaces in 1914.That concept was created by Frechet in his thesis in 1906. $\endgroup$
    – KCd
    Feb 11, 2021 at 8:36
  • 2
    $\begingroup$ The reference [2] does not say Hausdorff conceived the notion of metric spaces. It says he "created a theory of topological and metric spaces". $\endgroup$ Feb 11, 2021 at 13:15


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.