Who wrote up Banach's thesis? Sometime ago I read somewhere (and I don't remember where it was) that Stefan Banach--a highly creative and great mathematician--did not always write down his ideas.
Allegedly, he did not write his own thesis (but of course, all the mathematics in it came from him). Is that true? And is it known who wrote it then?
 A: In the Fall 1988 issue of the Mathematical Intelligencer there is an interview of Andrzej Turowicz who was a contemporary of Banach and Mazur. Here is one of the questions.
Q: Were all the Lvov mathematicians so reluctant to publish their results?A: No, it was a specialty of Mazur. Banach also left many of his results unpublished, but for a different reason. Banach turned out mathematical ideas so quickly that he should have had three secretaries to compose his papers. That was why Banach published only a small part of the theorems he invented. Not because he did not want to, but because all the time he had new ideas.
A: 
I am hesitant to write here, I delayed/procrastinated for long. I am not a historian, I simply was embedded in the Polish mathematical scene for over ten years, and since then I kept in personal touch with several of my Polish mathematical friends.

The notion of Banach assistant is not right. Banach had students and (mathematical) friends, including and especially younger friends. The most important among them was Stanisław Mazur who himself was a fantastically sharp mathematician. Stanisław Mazur truly disliked writing (editing) mathematics because he was doing it so well. For instance, Stanisław Mazur wrote (i.e. edited) the first paper by KS, who was about 30 years younger. However, Prof. Mazur didn't care to publish his own results. Prof. Kuratowski told me that Mazur was happy when someone else rediscovered and published Mazur's results. Mazur would say happily on such occasions: it (the results) had to be good enough if someone bothered to publish it.
Sometime in 1971-72 (or on a next occasion?), Aleksander Pełczyński (Olek) told me, when he had visited me in Ann Arbor (MI) a couple of times, that Banach's classic Theory of Linear Operators was written (i.e. edited) by Mazur.
Stefan Banach didn't care to edit his own research results. However, he did write academic and high-school texts extremely well. At least, this is my opinion based on my studying Banach's 2-volume Calculus monography on my own, when I was a high school student--I'd wake up way before my school day and would read for an hour or two. For a contrast, earlier I had gotten another text--famous--on mathematical analysis by a polytechnic professor. I stopped reading it very soon because it was too boring.

In many places around the world people like to stress how hard they work. It was often the opposite in Poland, especially among many Polish mathematicians. They were particular about being young, brilliant, and lazy. They would not say that they worked hard but that it was nothing, it just came to me at one moment, something like this. Ulam's autobiography illustrates my point. (On the other hand, a close friend of Ulam considers Rota's writing about Ulam as offensive, abusive, dishonest.)

A: In Stanisław Ulam's autobiography Adventures of a Mathematician you can find several references in that sense (mainly in the first Part) about the mathematicians at Lwów in that time, maybe the clearest one is on page 38:
"In general, the Lwów mathematicians were on the whole somewhat reluctant to publish. Was
it a sort of pose or a psychological block? I don't know. It especially affected Banach, Mazur, and myself, but not Kuratowski, for example."
A: When I was a student in Lvov in the 1970s, I heard many legends about Banach, so let me add a few points. Once Steinhaus was walking in a park, and he accidentally heard a conversation of two young people sitting on a bench. The words "Lebesgue integral" were pronounced. At that time very few
people in Lvov had heard of the Lebesgue integral. So Steinhaus was curious, and introduced himself... Banach was an engineering student at that time. (The story does not tell who the other person sitting on the bench was.)
According to the legend, Banach worked most of his time in the Scottish café. Students and colleagues joined him for conversation. (One of the results of this was the famous "Scottish book" of unsolved problems. Prizes were offered sometimes and recorded to the book together with the problems. For example, in the 1970s, when Per Enflo solved the "basis problem" from
the Scottish book, he won a prize, a live goose, which was delivered by Mazur). He used to write on the table cloth. The owner of the cafe never complained. At the end of the day, he changed the tablecloth for a new one. And he would sell the old one to students.
Banach drank a lot (and there are many stories about this, which I omit).
Frequently he was short of money, and had to drink in credit.
At some time, the debt grew large, and there was an argument with the owner of the Scottish café. Finally, the owner proposed that Banach writes a calculus textbook to make money to pay for his drinks. (Some version of the legend says this was suggested by students). Indeed, he wrote a calculus textbook :-) But I have never seen his high school textbooks.
The Scottish café still existed in the 1990s, but under a different name, and in the 1970s this was a simple cantina. Then, the rooms passed to
some financial institution.
P.S. Wikipedia, https://en.wikipedia.org/wiki/Scottish_Caf%C3%A9, has somewhat different details of doing math in the Scottish café, based on Ulam's recollections.
A: I also once heard such a story, but I have doubts it is literally true. What is an established fact is that Banach had an unusual start of his career. 
He was actually an engineering student (with a personal situation rather on the difficult end) and did math more or less as a hobby. By pure coincidence he met Hugo Steinhaus who was impressed. They worked together and published something together. Then Banach got a position at a university (Lvov) and then a doctorate (under Lomnicki [correction: while he was working for/in the group of Lomnicki, it appears Lomnicki was in no sense the director of his thesis; cf Magaret Friedland's answer]). So he got his doctorate under somewhat unusual circumstances and not following standard rules (though at that time, there were much less rules for doctorates then nowadays anyway). 
In that sense, it was likely not so clear when and how he should submit his thesis, and it seems very conceivable that he discussed this matter with various people and/or people close to him pressured/encouraged/helped him to do so. (Added: I see Francesco Polizzi made a comment sort of in this direction.) 
Regarding the "laziness": 
Not long after the time of his thesis he wrote a lot (including high-school textbooks). So, to attributed this to sheer laziness in a classical sense seems certainly odd. If anything I could imagine a certain uncertainty (and/or occupation with other matters) regarding how to proceed; 
or how to really write mathematics (not being trained as a mathematician). 
Yet, it is also well-documented that he and others worked a lot in cafés. Now, this could to some be taken as a sign of a 'lazy' life-style. But, well, not even this is so clear. 
For an overview of Banach's life http://www-history.mcs.st-andrews.ac.uk/Biographies/Banach.html 
A: Here is a quote from the article by Krzysztof Ciesielski:  On Stefan Banach and some of his results.  Banach J. Math. Anal.  1  (2007),  no. 1, 1–10.

There is a curious story how Banach got his Ph.D. He was being forced
  to write a Ph.D. paper and take the examinations, as he very quickly
  obtained many important results, but he kept saying that he was not
  ready and perhaps he would invent something more interesting. At last
  the university authorities became nervous. Somebody wrote down
  Banach’s remarks on some problems, and this was accepted as an
  excellent Ph.D. dissertation. But an exam was also required. One day
  Banach was accosted in the corridor and asked to go to a Dean’s room,
  as “some people have come and they want to know some mathematical
  details, and you will certainly be able to answer their questions”.
  Banach willingly answered the questions, not realising that he was
  just being examined by a special commission that had come to Lvov for
  this purpose.

It is true that Banach was mainly self-taught as a mathematician, although he attended some lectures by Stanislaw Zaremba at Jagiellonian University. By the way, engineering programs in the former Austro-Hungarian monarchy (including Lvov Polytechnics) required quite an intensive training in mathematics, although of course the latest developments (Lebesgue integral etc.) were not part of the curriculum.
Addendum 0: The above story is also related by Roman Kaluza in his biography of Banach. He heard it from Turowicz, who credits Nikodym as his source (he himself joined the department later, when Banach was already a professor). Well, on one hand, Nikodym was a friend of Banach and his early partner in mathematical discussions, but on the other hand, at the time of Banach's PhD, he was teaching high school in Krakow. (This point was made by Krzysztof Ciesielski in an email exchange with me.) 
Addendum 1: Banach's thesis, written in French (which he knew well and used before in publications) can be found here: 
http://kielich.amu.edu.pl/Stefan_Banach/pdf/oeuvres2/305.pdf
It was published in Fundamenta Mathematicae 3 (1922), pp.133-181,  and bears only Banach's name. The footnote says that it is a "Thesis presented in June 1920 at the Lvov University for obtaining the degree of the Doctor of Philosophy." 
On the first page there is a statement that maybe gives some  evidence of Banach's tendency to  wait until getting the best version of his results: ``Mr. Wilkosz and I have some results (which we propose to publish later) on operations whose domains are sets of Duhamelian functions(...)". There is no joint work with Wilkosz in the collected works of Banach...
Addendum 2: Some details brought up by other users need correction. First, Steinhaus met Banach and Nikodym in Krakow, where Banach grew up, not in Lvov. This is explicitly recorded in his "Memoirs and Notes", and somewhat less explicitly in the address he gave much later  at a session devoted to Banach: http://kielich.amu.edu.pl/Stefan_Banach/steinhaus63.html ("Planty" is a major green belt in the old city of Krakow; in Lvov there  were "Waly"). 
Second, Banach's PhD supervisor (only in the formal sense) was Steinhaus. Antoni Lomnicki held a chair of mathematics at the Lvov Polytechnics (not to be confused with the Lvov University), where Banach got his first position as an assistant (pre-PhD).
Addendum 3: Here is what Steinhaus wrote about working habits of Banach (see the link above):

His lectures were excellent; he never lost himself in particulars, he
  never covered the blackboard with numerous and complicated symbols. He
  did not care for verbal perfection; all manner of personal polish was
  alien for him and, throughout his life he retained, in his speech and
  manners, some characteristics of a Cracow street urchin. He found it
  very difficult to formulate his thoughts in writing. He used to write
  his manuscripts on loose sheets torn out of a notebook; when it was
  necessary to alter any parts of the text, he would simply cut out the
  superfluous parts and stick underneath a piece of clean paper, on
  which he would write the new version. Had it not been for the aid of
  his friends and assistants, Banach's first studies would have never
  got to any printing office." And also: "Banach could work at all times
  and everywhere. He was not used to comfort and he did not want any. A
  professor's earnings ought to have supplied all his needs amply. But
  his love of spending his life in cafes and a complete lack of
  bourgeois thrift and regularity in everyday affairs made him incur
  debts, and, finally, he found himself in a very difficult situation.
  In order to get out of it he started writing textbooks.

Addendum 4: This is based on information I received from Danuta Ciesielska, a Polish mathematician and a historian of mathematics (and my classmate from Krakow). The documents from the Lvov University are now split between the Lvov District Archive and Lvov City Archive, http://www.archives.gov.ua/Eng/Archives/ra13.php - Wayback Machine link (the documents of Polytechnics were transported to Wroclaw, Poland after 1945). The catalogs underwent major reorganization, which makes it quite difficult to find particular documents there. Besides the employees' folders, the documentation of PhD and habilitation proceedings is often found in the minutes of faculty meetings.  Regarding Banach's PhD, Ciesielska saw a letter from Steinhaus to dean Stanecki (dated September 28, 1920) asking him to set the date for Banach's doctoral exam, to which Stanecki replied that the date cannot be set before Messrs. Steinhaus and Zylinski (the committee members) evaluate the thesis. (Aside: Math Genealogy Project lists Kazimierz Twardowski as one of Banach's advisors. On the surface of it, this makes little sense, as  Twardowski was a philosopher and a logician; his expertise was far removed from what Banach worked on. However, as a professor of Lvov University, he was on the committee  and signed the papers.)
She also points out that in some institutions (e.g., Jagiellonian University in Krakow),   if a PhD thesis was published after the exam, the printed copy/journal offprint replaced the submitted manuscript/typescript. It is not clear if this was the case in Lvov.
A: I think this question is very subjective, speculative and gossipy, and I am surprised that it has not yet been criticized as not suitable for MO.
Unlike in mathematics, in history it is often enough to raise an unsubstantiated question in order to influence people's beliefs. It is very easy to spread rumours in history, and it is therefore important to provide good evidence for any suggestion that has to do with a historical fact.
What evidence do you have that Banach did not write his thesis, and what makes you think that the word 'lazy' is appropriate here? Would you call Hardy lazy because he only worked a couple of hours a day and spent the rest of his days reading about cricket? Would you call Grothendieck lazy because he did not write up his proof of Grothendieck-Riemann-Roch? Certainly not, because these people, just like Banach, were very prolific and influential mathematicians.
In a similar way, Rota's description of Ulam is historically unhelpful, and only illustrates the fact that Rota sometimes described people in rather arrogant terms (as he also did with Artin in Indiscrete Thoughts).
Please let us stick to the facts and not make MO a forum for speculative historical anecdotes. 
