In mathematics, especially functional analysis, a Fréchet algebra, named after Maurice René Fréchet, is an associative algebra
over the real or complex numbers that at the same time is also a (locally convex) Fréchet space.
The multiplication operation
is required to be jointly continuous.
is an increasing family[a] of seminorms for the topology of
, the joint continuity of multiplication is equivalent to there being a constant
[b] Fréchet algebras are also called B0-algebras.
-convex if there exists such a family of semi-norms for which
In that case, by rescaling the seminorms, we may also take
and the seminorms are said to be submultiplicative:
-convex Fréchet algebras may also be called Fréchet algebras.
[2] A Fréchet algebra may or may not have an identity element
is unital, we do not require that
We can drop the requirement for the algebra to be locally convex, but still a complete metric space.
In this case, the underlying space may be called a Fréchet space[12] or an F-space.
[13] If the requirement that the number of seminorms be countable is dropped, the algebra becomes locally convex (LC) or locally multiplicatively convex (LMC).
[14] A complete LMC algebra is called an Arens-Michael algebra.
[15] The question of whether all linear multiplicative functionals on an
-convex Frechet algebra are continuous is known as Michael's Conjecture.
[16] This conjecture is perhaps the most famous open problem in the theory of topological algebras.