  A subset S of a Riemannian manifold N is called extrinsically homogeneous if
S is an orbit of a subgroup of the isometry group of N. Thorbergsson proved the
remarkable result that every complete, connected, full, irreducible
isoparametric submanifold of a finite dimensional Euclidean space of rank at
least 3 is extrinsically homogeneous. This result, combined with results of
Palais-Terng and Dadok, finally classified irreducible isoparametric
submanifolds of a finite dimensional Euclidean space of rank at least 3. While
Thorbergsson's proof used Tits buildings, a simpler proof without using Tits
buildings was given by Olmos. The main purpose of this paper is to extend
Thorbergsson's result to the infinite dimensional case.
