[FOM] predicativism and functional analysis
Nik Weaver
nweaver at math.wustl.edu
Wed Feb 22 23:45:52 EST 2006
John Griesmer wrote (quoting me):
> > We have a name for the separable sequence space c_0. We
> > have a name for its (separable) dual l^1. We have a name for
> > l^infinity, the (nonseparable) dual of l^1. We have no name
> > for the dual of l^infinity.
>
> The dual of l^infinity is called M(beta(N)), where beta(N) is the
> Stone-Cech compactification of N (with the discrete topology).
Okay, that's fair. We certainly have a standard name for beta N,
so I have to agree that we have a standard name for the space of
regular Borel measures on beta N, namely M(beta N).
In fairness to my point, you would surely admit that c_0, l^1, and
l^infinity are everyday objects in functional analysis and M(beta N)
is not. In fifteen years as a functional analyst I don't think I've
every seen anyone use M(beta N) for any purpose.
You might also grant that my other examples are okay:
The compact operators on a Hilbert space: K(H); the dual of K(H),
the trace class operators: TC(H); the dual of TC(H), the bounded
operators: B(H); the dual of B(H): no standard name.
L^1[0,1]; its dual: L^infinity[0,1]; its dual: no standard name.
C[0,1]; its dual: M[0,1]; its dual: no standard name.
AE[0,1]; its dual: Lip_0[0,1]; its dual: no standard name.
where in each case the first dual that has no standard name is
also the first impredicative dual.
So my assertion
> all of the standard Banach spaces that functional analysts
> care about enough to have given them special names ... are
> in fact predicatively legitimate. If you look at the dual
> of such a space, you invariably find that it has a special
> name if and only if it is predicatively legitimate.
is not accurate. Actually there are plenty of exceptions in
one direction, where the dual of a named space has no special
name even though it is predicatively legitimate, just because
it's not so important. In the other direction there is one
example of a space, M(beta N), which is not predicatively
legitimate and is of marginal importance but does have a
special name.
Nik
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