When I was in grad school, I had a course in Banach spaces with Haskell Rosenthal. One day he said “We got the definition wrong.” It took a while to understand what he meant.
There’s nothing logically inconsistent about the definition of Banach spaces. What I believe he meant is that the definition is too broad to permit nice classification theorems.
I had intended to specialize in functional analysis in grad school, but my impression after taking that course was that researchers in the field, at least locally, were only interested in questions of the form “Does every Banach space have the property …” In my mind, this translated to “Can you construct a space so pathological that it lacks a property enjoyed by every space that anyone cares about?” This was not for me.
I ended up studying differential equations. I found it more interesting to use Banach spaces to prove theorems about PDEs than to study them for their own sake. From my perspective there was nothing wrong with their definition.
5 thoughts on “We got the definition wrong”
This feels a lot like the situation in commutative algebra (very strange properties of rings). In fact, I think functional analysis is to PDEs as commutative algebra is to algebraic/arithmetic geometry.
A bit like sets, one might say if in a particularly category-theoretic mood :-)
It makes me wonder what other situations the definition was wrong. Off the tlop of my head:
In electronics, the definition of capacitance is wrong. The right definition is the reciprocal of capacitance.
Tau advocates think the definition of pi is wrong.
Some are of the opinion that the definition of “prime number” is wrong, and it should include 1. (These people also think that the trivial “field” is a real field.)
Many physicists believe that electric charge has the wrong sign.
The gamma function is (I think) wrong; we should use the factorial function instead. Planck’s constant is wrong by a factor of 2 pi (or, if you prefer, tau).
It’s not just the shift-by-one that is wrong in the gamma function: a lot of formulæ come out better if you also scale by exp(γz) where γ is the Euler–Mascheroni constant.