[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.


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