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.