There are at least two theorems known as Weierstrass's theorem. The first states that the only hypercomplex number systems
with commutative multiplication
and addition are the algebra
with one unit such that 
and the Gaussian integers.
In harmonic analysis, let 
be any open set, and
let 
,
,
..., be a finite or infinite sequence in 
(possibly with repetitions) that has no accumulation
point in 
.
There exists an analytic function 
on 
whose zero set is precisely 
(Krantz 1999, p. 111). This is also sometimes known
as the Weierstrass product theorem.
