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
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