4 S. YAMAMURO

p o s i t i v e numbers a . and semi-norm maps p . € T ( l i n ) such t h a t q i s

e q u i v a l e n t t o (cup..) u ( a

0

p

0

) u . . . u ( a p ) . Thi s s e t f w i l l be c a l l e d

^ 1 1 2 2 n^n

t h e closure of F , and Y w i l l be s a i d t o be closed i f T = F .

Let T be a c a l i b r a t i o n fo r E and q be a semi-norm map on E . Then,

we pu t

Tq = {p u q : p € T} ,

which is again a calibration for E . When q € T , q is the smallest element

of T ; conversely, any calibration with the smallest element is in this form.

EXAMPLE 1. Let W be the family of all normed spaces. Then, each E € hi is

equipped with a norm XE . Therefore, A is a (semi-) norm map on W and the

set consisting only of X is a calibration for W .

EXAMPLE 2. Let E and F be locally convex spaces and T be a calibration

for the product space E x F . For each p € T , we put

p£(x) = p[(x,0)] and Pp(y) = p[(0,y)]

for every (x,y) € E x F . Then, p is a semi-norm map on the family

E={E,F} and V is a calibration for E . Similarly, we can produce a

calibration for an arbitrary family of locally convex spaces.

EXAMPLE 3. Let r be a calibration for E which consists of two spaces E

and F . Assume that there is a map O of r„ into rr . Then, the set T

F E a

which consists of all p € T whose E-components are replaced by pr u j(p) is

again a calibration for E . This will be called a calibration generated by

T and a .

Remark In the theory of differentiation in normed spaces, essential tools are

inequalities among quantities measured by norms, rather than the topology of the

space. In the following argument on locally convex spaces, a suitably chosen

calibration will replace the norm and we shall try to get corresponding