The Corner

Absolutely the Last Hilbert Space Post

I don’t much care for breezy optimism, in higher math nor anywhere else; but this email is from a dear friend (whose first language was not English)  and fine mathematician, so I suppose I ought to post it:

“Hilbert space still allows you to hear.  Hilbert space as defined by Kolmogorov and Fomin’s textbook on functional analysis contains this example:

“All the infinite sequences of complex numbers c_n,  1 <= n <= infinity, on assumption that the sum of their squares of moduli is __convergent__ !

“It means that the actual vectors of Hilbert space are ‘concentrated’ in the finite numbers ‘n’, so that the fraction (energy, probablility) related to very large ‘n’ is negligibly small.

“So, while the Hilbert space is formally infinitely-dimensional, real life in them plays out in some finite number of dimensions.  The latter may grow from one problem to the other. 

“Never in my professionl life have I encounered the case where essential physics required more than one million of dimensions (actually 1000 with respect to x-coordinate and 1000 with respect to y-coordinate, while evolution with respect to z-coordinate and with respect to time was governed by the wave equation itself).  Even then the main ‘energy, probability’ was concentrated in around 100 x 100 = 10,000 really strong components, while the rest of the components served for the slight refinement of interference phenomena.

“Do not make formal sign of infinity lure you into despair.  Life is short in time and in space, including Hilbert space!”

[Derb]  Layman’s translation:  If a tree falls in the forest, in a universe of infinitely many spatial dimensions (as opposed to our niggardly three), it _will_ make a noise sufficient to wake any hamsters napping in the vicinity.

I’m sure that we are all glad to know that a million dimensions suffices for all practical purposes.

John Derbyshire — Mr. Derbyshire is a former contributing editor of National Review.

Most Popular

Film & TV

A Sad Finale

Spoilers Ahead. Look, I share David’s love of Game of Thrones. But I thought the finale was largely a bust, for failings David mostly acknowledges in passing (but does not allow to dampen his ardor). The problems with the finale were largely the problems of this entire season. Characters that had been ... Read More
Politics & Policy

The Great Misdirection

The House Democrats are frustrated, very frustrated. They’ve gotten themselves entangled in procedural disputes with the Trump administration that no one particularly cares about and that might be litigated for a very long time. A Washington Post report over the weekend spelled out how stymied Democrats ... Read More

Australia’s Voters Reject Leftist Ideas

Hell hath no fury greater than left-wingers who lose an election in a surprise upset. Think Brexit in 2016. Think Trump’s victory the same year. Now add Australia. Conservative prime minister Scott Morrison shocked pollsters and pundits alike with his victory on Saturday, and the reaction has been brutal ... Read More
NR Webathon

We’ve Had Bill Barr’s Back

One of the more dismaying features of the national political debate lately is how casually and cynically Attorney General Bill Barr has been smeared. He is routinely compared to Roy Cohn on a cable-TV program that prides itself on assembling the most thoughtful and plugged-in political analysts and ... Read More
Film & TV

Game of Thrones: A Father’s Legacy Endures

Warning! If you don't want to read any spoilers from last night's series finale of Game of Thrones, stop reading. Right now. There is a lot to unpack about the Thrones finale, and I fully understand many of the criticisms I read on Twitter and elsewhere. Yes, the show was compressed. Yes, there were moments ... Read More