Fréchet algebra

In mathematics, a Fréchet algebra (after Maurice René Fréchet) is a topological algebra, 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.^[1]

References

^ Ernest A. Michael. "Locally multiplicatively convex topological algebras". Mem. Amer. Math. Soc., 11, 1953.

Zelazko, W. (2001), "Fréchet algebra", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=f/f110170
L. Waelbroeck, Topological Vector Spaces and Algebras, Lect. Notes Math. 230, Berlin-Heidelberg-New York, Springer-Verlag, 1971.
E. A. Michael, Locally multiplicatively-convex topological algebras, Mem. Amer. Math. Soc. 11 (1952), 1-79.
B. Mitiagin, S. Rolewicz, and W. Zelazko, Entire functions in $B_0$ algebras, Stud. Math. 21 (1961), 291-306.