A theory of differentiation stands on a theory of linear maps, and the
theory of linear maps on locally convex spaces which we have today is not suit-
able as the basis for a good theory of differentiation, because of the following
two facts in the algebra L(E) of all continuous linear maps of a locally con-
vex space E into itself:
(1) If the multiplication in L(E) is jointly continuous, then E is
(2) If L(E) is a continuous inverse algebra, then E is normable.
The first fact prevents us from obtaining good chain rules, and the second
fact is the reason why we do not have good inverse mapping theorems.
We know that any Banach algebra has jointly continuous multiplication and
is a continuous inverse algebra, i.e., the set of all isomorphisms is open and
the inverse operation is continuous on this open set (, p.87). Therefore,
the above facts disappear if we replace L(E) with a subalgebra which can be
made into a normed algebra.
The notion of bounded elements, introduced by Allan ,seems to be suit-
able for this purpose. Robert T. Moore  has developed such a theory in the
course of his study on one-parameter semi-groups, numerical ranges and other
problems. John Giles and his co-workers [7, 8, 10] have developed a theory of
this type further. We owe to Moore the terminology "calibration for E" which
means a set T of continuous semi-norms on E which induces the topology of
E . Instead of L(E) , he attached to E the set L r(E) of all linear maps
Received by the editors: July 2, 1976; in revised form: February 23, 1977.