Paul Erdos’s Two-Line Functional Analysis Proof

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?

Answer

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

Attribution
Source : Link , Question Author : Emily , Answer Author : JSchlather

Leave a Comment