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]