Uniform approximation by products #
We show that if X, Y
are compact Hausdorff spaces with X
profinite, then any continuous function
on X × Y
valued in a ring (with a uniform structure) can be uniformly approximated by finite
sums of functions of the form f x * g y
.
A continuous function on X × Y
, taking values in an R
-module with a uniform structure,
can be uniformly approximated by sums of functions of the form (x, y) ↦ f x • g y
.
Note that no continuity properties are assumed either for multiplication on R
, or for the scalar
multiplication of R
on V
.
A continuous function on X × Y
, taking values in a ring R
equipped with a uniformity
compatible with addition, can be uniformly approximated by sums of functions of the form
(x, y) ↦ f x * g y
.
Note that no assumption is needed relating the multiplication on R
to the uniformity.