TITLE: Paul Erdos's Two-Line Functional Analysis Proof
QUESTION [85 upvotes]: Legends hold that once upon a time, some mathematicians were rather pleased about a 30-ish page result in functional analysis. Paul Erdos, upon learning of the problem, spent ten or so minutes thinking about the original problem, and came up with a two-line proof.
I believe I read about this first in the biography The Man Who Loved Only Numbers, and it seems as though the internet maintains this legend (q.v. here, here, here).
This seems extraordinary, which leads me to some skepticism. I cannot seem to find any other reference to the actual proof or the problem. Is this simply an urban legend? Did it actually happen? What was the problem, and what was the original 30-page result?

REPLY [5 votes]: There is another well documented story in which Erdos went to a seminar where an open problem in the topology of infinite-dimensional spaces was stated. I think it was something about determining the dimension (for a particular definition of dimension) of the rational points in an infinite-dimensional Hilbert cube, and that the problem or the seminar was by Hurewicz.   Erdos did not solve it in seconds, but he did so very quickly on the spot, while the seminar was going on. The solution was by his own combinatorial methods.