In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Fréchet algebra (after Maurice René Fréchet

Maurice René Fréchet

Maurice Fréchet was a French mathematician. He made major contributions to the topology of point sets and introduced the entire concept of metric spaces. He also made several important contributions to the field of statistics and probability, as well as calculus...

) is a topological algebra

Topological algebra

In mathematics, a topological algebra A over a topological field K is a topological vector space together with a continuous multiplication\cdot :A\times A \longrightarrow A\longmapsto a\cdot bthat makes it an algebra over K...

, in which the topology is given by a countable family of submultiplicative seminorms:

p(fg) ≤ p(f)p(g),

and the algebra is complete.

For example, A can be equal to C(C), the algebra of all continuous functions on
the complex plane C, or to the algebra Hol(C) of holomorphic functions on C, both
equipped with the topology of uniform convergence on compact sets. Perhaps the
most famous, still open problem of the theory of topological algebras is whether all
linear-multiplicative functional on a Frechet algebra are continuous. The statement that this be the case is known as Michael's Conjecture.