[FOM] predicativism and functional analysis

Nik Weaver nweaver at math.wustl.edu
Wed Feb 22 03:29:08 EST 2006

I pointed out a fact that I find quite remarkable: although
in general one cannot predicatively form the dual of a Banach
space, all of the standard Banach spaces that functional
analysts care about enough to have given them special names
(sequence spaces, spaces of continuous functions, spaces of
measurable functions, spaces of bounded operators, etc.) 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.  Thus,
the dual of l^1 is l^infinity (predicatively legitimate).
The dual of l^infinity has no standard name (predicatively

Harvey Friedman wrote:

> Set theorist: NAMES? The only names core mathematicians want
> to hear about are the names of the great theorems and the great
> mathematicians who prove them and of the next prize winners and
> of the next hires.

Not sure what this has to do with my point.

Speaking as a core mathematician, I find your vision of what
we're like rather sad.

> Set theorist: Name, shame.


> > We have no name for the dual of B(H).  K(H) and TC(H) are
> > separable; B(H) is not.
> Set theorist: WHAT? No name now?

Are you sure you understand my point?  Functional analysts have
a standard name for the space of bounded operators on a Hilbert
space.  They call it B(H).  They do not have a standard name for
the dual of this space.  B(H) is predicatively legitimate; its
dual is not.

> > 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.
> Set theorist: Gee, I wonder if that could be because you are doing
> the naming and you are trying to avoid (ban) impredicativity?

I really don't think you understand my point.  No, I'm not
personally the one doing the naming.  The functional analysis
community as a whole does this.  All of the examples I gave
were of spaces whose notations had been standardized long
before I was born.


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