S . YAMAMURO
of E int o i t s e l f such tha t
||u||r = sup {p[u(x)] : p(x) 5 1, p€r} + » ,
which defines a norm of the algebra LT5_,(E) .
•Dl
It turned out that this algebra was not big enough; Moore, in an unpublish-
ed note, pointed out that there was a linear map of very important nature (See,
§3. Example 2) which did not belong to LR„(E) for any choice of calibration
T for E .
This defect can be remedied i f we regard two spaces E and F , even i f
E = F , with differen t calibration s as differen t objects . In [25] we have
presented such a method based on calibration s for E x F shown in Example 2 in
§1 below. To each pai r (E,F) , we have attached the space L _,(E,F) of a l l
linea r maps u of E int o F such tha t
||u||r = sup {p
p
[u(x) ] : p
£
(x ) 5 1, p ^ } + °° ,
which defines a norm on LRp(E,F) . Then, any continuous linear maps of E
into F belongs to L (E,F) for some calibration T for E x F (See (3.4)).
In other words, when we have a continuous linear map u : E - F , we can put it
in some LTJT1(E,F) , although LDT,(E,F) itself may be small (See (3.2)).
Taking a calibration T for E x F is nothing but choosing calibrations
Tp and Tr for E and F respectively and defining a map a : TF - TF .
This map transforms p in rp to its source p in T and then to itsE-
component pp .
Conversely, if we have a map a which maps continuous semi-norms on F
into continuous semi-norms on E , then we can construct a calibration for
E x F for which O is the map considered above. In practice, calibrations
appear in this way.
The idea of taking all maps between calibrations and introducing a pseudo-
topology in L(E,F) is due to Marinescu [14, 15]. This "Marinescu structure"
of L(E,F) has been studied further by H.H. Keller, whose book [11.p. 46]
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