  In 1952, Michael posed a question about the functional continuity of
commutative Frechet algebras in his memoir, known as Michael problem in the
literature. We settle this in the affirmative along with its various equivalent
forms, even for the non-commutative case. Indeed, we continue our recent works,
and develop two approaches to directly attack these problems. The first
approach is to show that the test case for this problem, the Frechet algebra of
all entire functions on the Banach space of all bounded complex sequences, is,
in fact, a Frechet algebra of all complex formal power series in one
indeterminate, if there exists a discontinuous character. In the second
approach, the existence of a discontinuous character would allow us to generate
other Frechet algebra topology, inequivalent to the usual Frechet algebra
topology, by applying the method of Read (he used this method to show that the
famous Singer-Wermer conjecture (1955) fails in the Frechet case). In both the
approaches, an important tool is a topological version of the (symmetric)
tensor algebra over a Banach space; the elementary, but crucial, idea is to
express the test algebra as a weighted Frechet symmetric algebra over the
Banach space of all absolutely summable complex sequences. Several
mathematicians have worked on two problems of Michael since 1952, giving
affirmative solutions for special classes of Frechet algebras under various
conditions, or discussing various test cases, or discussing various approaches,
or discussing various other equivalent forms, or deriving other important
automatic continuity results such as the (non-)uniqueness of the Frechet
algebra topology for certain commutative Frechet algebras by alternate,
difficult or lengthy methods. We summarize effects of our affirmative solutions
on these attempts in addition to giving various new (important) applications in
automatic continuity theory.
