[FOM] predicativism and functional analysis

[Nik Weaver]

This is a follow-up to a previous post of mine,
http://www.cs.nyu.edu/pipermail/fom/2006-February/009778.html
in which I argued that Cantorian set theory fits normal
mathematical practice rather poorly, because core mathematics
takes place only at the very bottom of the cumulative hierarchy.

Here I want to illustrate the point that there is a remarkably
exact fit between predicativism and core mathematics.  I would
even say a *stunningly* exact fit!

(I hasten to add that this is only weak evidence for the actual
correctness of predicativism, although I do think it is some
evidence in favor of correctness.)

I'll use functional analysis as my illustration.  Here the objects
of study have no a priori cardinality restrictions, so the field
is not as it were "automatically" predicative.

At least in my interpretation of predicativism, which I call
"conceptualism", it is possible to treat things like the real
line and the power set of N essentially as proper classes.
Let me simply use the term "proper class" to refer to such
things.  (Everything I'm writing here is made precise in my
paper "Analysis in J_2".)

What constructions are possible with proper classes?  Well,
we can form the cartesian product of two classes.  So we can
get spaces like R^n as proper classes.  Moreover, we can form
the cartesian product of any set of proper classes.  More
precisely, if S is a class contained in the cartesian product
A x B where A is a class and B is a set, then we can form the
cartesian product of the classes S_b = {a in A: (a,b) in S}
over b in B.  So we can even get R^omega as a proper class,
and sitting inside it l^p sequence spaces.

It's not hard to see that one can recover all of the standard
Banach spaces as proper classes.  What about standard Banach
space constructions?  The dual space construction particularly
concerned me because this is such an important construction in
functional analysis.  On its face the construction is problematic
because the elements of the dual of E are the continuous linear
maps from E to R.  (Let's assume the scalar field is real.)
However, any function from one proper class to another proper
class must itself be a proper class of ordered pairs.  So we
can't talk about the class of all continuous linear functionals
because a proper class can't be an element of another class.
We can talk about individual continuous linear functionals but
not the Banach space they constitute.

This problem is easily resolved if E is separable.  In that case
there is a countable dense subset and every continuous linear
functional is determined by its values on that subset, so we
can effectively restrict these functionals to the subset; this
renders them sets of ordered pairs and we can then form the class
of all such functions.

Second duals are trickier.  This bothered me for a while.  It
seemed that second duals are a critical test of predicativism
because second dual techniques really are fundamental in
functional analysis, but according to the above reasoning one
can't generally form the second dual even of a separable Banach
space because its first dual need not be separable.

What I finally realized is that standard second dual techniques
don't actually make use of the second dual as a Banach space.
They only use individual elements of the second dual, which is
predicatively okay!  For example, the most basic theorem about
the second dual says that the unit ball of E is weak* dense in
the unit ball of E''.  That could be rephrased as saying that
every norm one vector in E'' can be approximated by norm one
vectors in E, which is a statement about individual elements
of E''.

In ZFC, starting with any Banach space E you can successively
form E', E'', E''', etc.  However, in practice functional
analysts almost never use any dual beyond the second, and
they generally use the second dual only to the extent that
predicativity allows.  That is an *exact fit*.

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.

We have a name for the spaces of compact operators on a separable
Hilbert space: K(H).  We have a name for its dual, the space of
trace-class operators: TC(H).  We have a name for the dual of
TC(H), the space of bounded operators on a Hilbert space: B(H).
We have no name for the dual of B(H).  K(H) and TC(H) are
separable; B(H) is not.

We have a name for C[0,1] (separable).  We have a name for its
dual, M[0,1] (non-separable).  We have no name for the dual of
M[0,1].

It goes on and on.  If you pick a standard Banach space and
start taking duals, invariably the first one that doesn't have
a standard name is the first one that's impredicative.  This
is a *stunningly exact fit* between predicativism and core
mathematics.

Nik Weaver
Math Dept.
Washington University
St. Louis, MO 63130 USA
nweaver at math.wustl.edu
http://www.math.wustl.edu/~nweaver