The
small gain theorem still holds even if *D* and *M* are infinite dimensional. This is
summarized as the following corollary.

q**Corollary:** The following statements
are equivalent:

(i) The
system is well-posed and internally stable for all *D** *Î*H*¥ with ||*D*||¥ £1/g .

(ii) The
system is well-posed and internally stable for all *D** *Î*RH*¥ with ||*D*||¥ £1/g .

(iii) The
system is well-posed and internally stable for all *D** *Î**R***qxp* with ||*D*|| £1/g .

(iv) ||*M*||¥ £ g

It can be
shown that the small gain condition is sufficient to guarantee internal
stability even if *D* is a nonlinear and time varying “stable” operator
with an appropriately defined stability notion, see Desoer and
Vidyasagar [1975].

Corollary