[FOM] predicativism and functional analysis
griesmer at math.ohio-state.edu
Wed Feb 22 16:32:04 EST 2006
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
(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
(Neil Hindman has shown that we cannot necessarily take A=B above.)
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