The so-called "Hilbert space" is named after mathematician David Hilbert. Later, this was generalized into "Banach spaces" by Stefan Banach.

My understanding is that Hilbert was German and Banach was Polish, and there did not appear to be any "major" connection between them (that is, no more than between two "random" European mathematicians, although this was a very small circle at the time). Yet there is a fairly strong connection between Hilbert's work and Banach's work.

How did Banach manage to take off on Hilbert's work without knowing him well? (E.g. Banach seems to have been much closer to Hugo Steinhaus of the Banach-Steinhaus Theorem.) Or did the two work together/know each other better than I have given them credit for?


3 Answers 3


It is worth noting that the abstract definition of a Hilbert space (as a complete inner-product space) is not due to Hilbert. Weyl recounts the history in his memorial essay, "David Hilbert and His Mathematical Work" (Bull. Amer. Math. Soc. v.50 p.612--654). In his work on integral equations, Hilbert investigated only one particular Hilbert space: the space of square-summable infinite sequences. He did not use Lebesgue integration; only later did Riesz and Fischer show the equivalence with Lebesgue square-integrable functions. Weyl adds:

I mention these details because the historic order of events may have fallen into oblivion with many of our younger mathematicians, for whom Hilbert space has assumed that abstract connotation which no longer distinguishes between the two realizations...

(There is also an apocryphal story that Hilbert attended a lecture, and came up at the end to ask the speaker, "What is a Hilbert space?")

Banach by contrast gave the abstract formulation of Banach spaces in his dissertation, along with his motivation:

This present work has the object of establishing certain theorems that hold in several different branches of mathematics, which will be specified later. However, in order to avoid proving these theorems for each branch individually, which would be very wearisome, I have chosen a different way, which is this: I consider in a general way sets of elements for which I postulate certain properties. From these I deduce theorems and then I prove for each separate branch of mathematics that the postulates adopted are true of it.

In other words, Banach is seeking economy of proof via the axiomatic method. His motivation is thus entirely different from Hilbert's.

To return to your original question: I have not been able to uncover any personal connection between Hilbert and Banach. The name "Banach" does not appear in the index of Constance Reid's biography Hilbert; the MacTutor entry for Hilbert does not contain "Banach", and the MacTutor entry for Banach contains only once occurrence of Hilbert, where it notes that Banach's work "generalised the contributions made by Volterra, Fredholm and Hilbert on integral equations".

However, that one sentence is probably sufficient explanation. Hilbert did his work on integral equations in the early 1900s, and it was soon developed further by Riesz, Fischer, Schmidt, and others. Banach's dissertation was written in 1920. It is hardly surprising that entering this field, Banach would pay close attention to relevant published work by one of the foremost mathematicians of the day.

  • 2
    $\begingroup$ To stress that Hilbert did in fact not study "his" spaces in an abstract manner is really a good point to make. $\endgroup$
    – quid
    Oct 31, 2014 at 0:42
  • $\begingroup$ ...although I think this is an interesting read, and it's worth preserving, it does not answer the question. Can you expand your answer to actually deal with the main question from the OP? $\endgroup$
    – Danu
    Nov 2, 2014 at 1:37
  • $\begingroup$ OK, I've added two paragraphs. $\endgroup$ Nov 2, 2014 at 5:28
  • $\begingroup$ +1 published work. This answers the question "How did Banach manage to take off on Hilbert's work without knowing him well?" Indeed, most of my own work is based on published work of those I never met. $\endgroup$ Mar 6, 2022 at 14:16

As far as I know there is no particular connection between Hilbert and Banach. Of course, Hilbert being one of the most dominant mathematicians of the time his influence was wide spread.

It would however also be wrong to consider first Hilbert then Banach as a direct succession. There were various influences and contributors in the development of what are now Banach spaces. [Indeed, the notion was almost in parallel introduced by others, too, Wiener in particular. (Banach was the one that made most out of it and rightly got the "name credit")] Other names one could mention besides Hilbert include Fredholm, Riesz, Fischer, Fréchet, Lebesgue.

To wit, the chronology in Pietsch's History of Banach spaces and linear operators has 12 entries (starting 1902) before Banachs thesis in 1920.

In this context it might be also noteworthy that Banach visited Paris in 1924-25.


There is no known record of any personal encounter between Banach and Hilbert. But the not-so-random connection between the two was Hugo Steinhaus (Banach's discoverer, and later collaborator and colleague), who was a PhD student of Hilbert in Göttingen. Steinhaus's thesis, titled Neue Anwendungen des Dirichlet'schen Prinzips, defended in 1911, was still rather traditional in its approach to variational problems for second-order partial differential equations.

On the other hand, Banach's PhD thesis O operacjach na zbiorach abstrakcyjnych z zastosowaniami do równań cał kowych [On operations on abstract sets with applications to integral equations] defended in Lwów in 1920, introduced fundamental notions and properties of linear normed complete spaces (in an axiomatic way), and applied them to integral operators defined by kernels. The actual thesis of Banach and its defense became stuff of legends, but at least there is a publication based on the thesis, S. Banach, Sur les opérations dans les ensembles abstracts et leur application aux équations intégrales}, Fundamenta Mathematicae 3 (1922), pp. 133-181. In the introduction Banach mentions preceding work on ``functional operations'' by Volterra, Fréchet, Hadamard, F. Riesz, Pincherle, Steinhaus, Weyl, Lebesgue and others. He particularly credits the works of Hilbert, which according to him enabled treatment of (the spaces of) square-integrable functions, not only smooth functions. This is also evidence that Banach studied works of Hilbert before visiting Paris.

Moreover, in 1917 Banach and Steinhaus both lived in Krakow and took part in meetings of an informal mathematical society. Other members of this group were mathematicians Wlodzimierz Stozek, Wladyslaw Slebodzinski, Leon Chwistek (also a philosopher and a painter) and a physicist Jan Norbert Kroo, all of whom spent some time studying in Göttingen.


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