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.
Examples
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.
Open problems
Perhaps themost 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]
Notes
References
- Zelazko, W. (2001), "Fréchet algebra", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Waelbroeck, Lucien (1971), Topological Vector Spaces and Algebras, Lecture Notes in Mathematics 230, Berlin: Springer-Verlag, doi:10.1007/BFb0061234, ISBN 978-3-540-05650-8, MR 0467234.
- Michael, Ernest A. (1952), Locally Multiplicatively-Convex Topological Algebras, Memoirs of the American Mathematical Society 11, MR 0051444.
- Mitiagin, B.; Rolewicz, S.; Żelazko, W. (1962), "Entire functions in B0-algebras", Studia Mathematica 21: 291–306, MR 0144222.