DIFFERENTIATION
3
contains a brie f account on hi s r e s u l t s .
§1. CALIBRATIONS
A calibratio n for a locall y convex space E i s a se t of continuous semi-
norms which induces the topology of E . The set P(E) of a l l continuous semi-
norms on E i s obviously the larges t calibratio n for E .
Let E = {E : a 1} be an indexed family of locall y convex spaces. Some
of the spaces in E may be equal, i . e . there may be different a and 3 in
I for which E = Eft as locall y convex spaces. A semi-norm map on E i s a
map p defined on I whose value p
p
a t a I belongs to P(E ) . We c a l l
a
a set T of semi-norm maps on E a calibration for E if, for each a I ,
the set
= {p£ : a I 1}
a
is a calibration for E . We shall also say that E is a T*-family.
In order to simplify the notations,we shall omit the indicies, i.e. we
shall write E , pE and TE instead of E , pp and TE respectively.
a a
Therefore, when we say that E and F are members of E or E, F E , it
means that there are a, 3 I such that E = E and F = E0 .
ot p
For two semi-norm maps p and q on E , we write p - q i f p
E
- qF
for each E E , which means tha t p
F
(x) q
F
(x) for a l l x E . Then,
p u q = max (p,q) and p n q = min (p,q) are well-defined.
For a semi-norm map p on E and a positiv e number a , we define ap as
the semi-norm map on E such tha t (otp)p = ap
p
for each E E .
The semi-norm maps p and q on E are said to be equivalent i f there
exis t positiv e numbers a and 3 such tha t aq p 5 3q .
Let F be a calibratio n for E . Then, we s h a l l denote by f the set of
a l l semi-norm maps q which satisf y the following condition: there exis t
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