[FOM] predicativism and functional analysis

[griesmer@math.ohio-state.edu]

Nik Weaver wrote:

> 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).  Beta(N) is
an important object in functional analysis, topological dynamics, and
Ramsey theory; its structure has far-reaching combinatorial consequences. 
For instance, one may use information about the topological and algebraic
structure of beta(N) to conclude the following:  If N is partitioned into
finitely many classes, one cell of the partition will contain each of the
following:

(i)   Arithmetic progressions of every finite length.
(ii)  Geometric progressions of every finite length.
(iii) An infinite set A and every finite sum with distinct summands from A.
(iv)  An infinite set B and every finite product with distinct factors
from B.

(Neil Hindman has shown that we cannot necessarily take A=B above.)

-John Griesmer