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Re: Today's jaw dropper



Date: Thu, 09 Oct 1997 18:16:17 EDT
From: twayburn@juno.com (Thomas L Wayburn)


On Thu, 9 Oct 97 14:31:33 CDT PPARKER@TWSUVM.UC.TWSU.EDU writes:
Date: Tue, 7 Oct 1997 15:35:13 -0700
From: Leigh Palmer <palmer@sfu.ca>

Even after Schwartz's work one of my references says
the Dirac delta function was *not* considered a respectable
function by mathematicians, but they admitted the use of a symbol
with all the same properties. Physicists persist in this "error"
with considerable profit, however.

If Dirac's delta is a function, then heat is a substance.
******************************************************************
The generalized fns are in the dual space of L2 . I guess that makes
them functions and the statement undecided.

Some generalized functions (Schwartz distributions) are in the dual of L^2,
but most aren't: they're in the dual of C-infinity with compact supports.
But elements of L^2 aren't functions (they're equivalence classes of
functions), so even there it doesn't work. And, finally, Dirac's delta
doesn't belong to the dual of L^2. In fact, it wouldn't even if the members
of L^2 were functions, because those functions that represent elements of
L^2 need not be defined everywhere, so some aren't defined at zero, so the
Dirac delta can't be evaluated on them.
By the way, the Dirac delta was "legitimized" by Gelfand, Mikusinsky, et
al. long before Schwartz (15 years or so). But Schwartz wrote a la Bourbaki
and the others wrote more traditionally. So Schwartz got the "popular"
credit, even though the real experts knew better.
Even more BTW, the work of Rosinger and Columbeau has finally "legitimized"
powers of the delta and all those other nonlinerar functionals that have been
wanted in theoretical physics (esp. QFT) for years. I suspect this will be
"discovered" eventually when a PC exposition appears.
Finally, at the very bottom, it's about understanding. Physicists have
definitions by which heat is not a substance. Mathematicians have them by
which the delta is not a function. Both claim that proper appreciation is
necessary for correct understanding and intellectual progress. So it's
high irony to find one who insists on correct usage of "heat" being smugly
dismissive about correct usage of "function."

---------------------------------------------
Phil Parker pparker@twsuvm.uc.twsu.edu
Random quote for this second:
Don't panic!---"The Hitchhiker's Guide to the Galaxy"