Paul Erdos's Two-Line Functional Analysis Proof

I sent Professor Purdy an email. I asked him what he recalled about the incident. With his permission I've copied his correspondence below.

Dear Jacob,

Yes, I was there, and I'm the one who told the story to Paul Hoffman, who then included it in his book "The man who loved only numbers."

The 30 page proof was written by Jack Bryan just before Erdos came to visit, at Texas A & M University. The problem was written on the blackboard in the mathematics lounge and Erdos saw it and asked "Is that a problem?" I told him yes, and he went over and wrote a two line proof on the blackboard. It's the most incredible thing I ever witnessed, and that's why I told Paul Hoffman the story. Ron Graham told Hoffman to talk to me because I knew Erdos well.

Much later it became obvious that Erdos loved this story, and I asked him how he did it without knowing the subject. He said, smiling, "Oh, I was a good student at school!"

I have come to realize also that Erdos was one of those mathematicians who could tackle an unknown area as if he knew it. He also prized this ability in others. He once demonstrated to me that Fred Galvin had this ability. Fred was at the board and Paul asked him a question, and he answered it, and he asked Fred "Had you seen these things before", and Fred answered that he hadn't, and Erdos turned to me and said "See!"

George

Yes, you are right. It was Jack Bryant, not Jack Brian. He might have retired by now. By the way, even before Erdos came to town, it was generally agreed that there must be a proof that was shorter than 30 pages, but not a two liner! Professor Don Allen talked to Erdos the next day to see if he could help with the research problem that had generated this result, but he reported to us later that unfortunately he couldn't. Don would doubtless remember the statement and proof of the two liner.

Professor George Purdy University of Cincinnati [email protected]

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.