those functions having compact support in I . We say that a linear sub-
space F c H is a k-semilocal subspace if it is stable under multiplication
by functions from Ck(I) , i.e. from n C (I) and f F it follows that
nf F. The most important examples of k-semilocal spaces we have in mind
are the Sobolev spaces of sections of direct integrals of Hilbert spaces
continuously embedded in H (see Section 5 ) .
Assume F is a k-semilocal subspace of H . We define F = {f F | the
support of f is a compact subset of I }, F = {f G L* (I;H) | if
n C (I) , then nf G F} . Clearly F is a k-semilocal subspace of H and
F _ is a linear subspace of Hn = Ln (I;H) which is stable under mul-
loc loc
tiplication by functions from C (I) ( we say that F is a k-semilocal
subspace of H. ) . But now we have more . namelJ y F and F. , are stable
lo c * c lo c
la -
under multiplication by any function from C (I) .
We say that a mapping T : F - H is local if , whenever J c I is open,
f1 , f2 G F and f1|J = f2|j , we have (Tf1)|j = (Tf2)|J . If T is (p-linear
(i.e. linear for the structures of complex vector spaces ) , then this is
equivalent to supp Tf c supp f for all f F . We say that T is k-ultralo-
cal if it is dJ-linear and Tnf = nTf for all n Ck(I), f G F. Remark
that if T is k-ultralocal . then it is local . TF c F and Tnf = nTf for
c c
all n G Ck(I) and f G F .
Assume that T : F * H is local . Then there is a unique mapping
T : F- . - H. . which is local and coincides with T on F ( if f C f, and
lo c lo c lo c
J c I is open with compact closure in I , choose n C (I) such that
n(t) = 1 in a neighbourhood of J and define (Tf)|j = (Tnf)|J.). Moreover ,
°° ^ lr
if T is k-ultralocal , then one has Tnf = nTf for each n G C (I) and all
f G F, ^ . From now on we shall denote the extension T by the same let-
loc °
ter T .
If G c H is a linear subspace . we define its k-semilocal closure
c 9
to be the smallest k-semilocal subspace of H that contains G . Clearly it
is equal to the linear subspace generated by elements of the form nf with
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