5 CURRENTS ON AN ORIENTED MANIFOLD 247
where the bH (resp. the #i_H) are the coefficients of p (resp. a) relative to the
canonical basis of the ^(U)-module <%(U) (resp. <^,0)(U)), and in the sum-
mation I = {1, 2,...,»} and H runs through all subsets of n — p elements of I.
Then we have to show that each of the linear mappings av _H i-» f £H(x)aI_H(jc) dx
is continuous on each of the Banach spaces «#T(U; K), where K is any compact
subset of U; and this follows from (13.13) because each of the functions bH is
locally integrable.
Let T0 be the p-current so defined. If we denote by <^M_PJ ioc(X) the vector
space of locally integrable differential (n — p)-forms on X, then we have a
linear mapping jSh->T^ of <fa_pf ,OC(X) into ^(0)(X). From (13.14.4) it follows
immediately that the kernel of this mapping is the subspace of negligible
(n — p)-forms. Since the support of a Lebesgue measure on X is the whole of
X, the restriction of the mapping jSh->T^ to the space ^^(X) of continuous
differential (n — p)-forms is injective, so that such a form may be identified
with a p-current of order 0. Under this identification, the notions of support
are the same for the continuous (n — p)-form p and the p-current T^ with
which we have identified it. For, by reducing as above to the case where X is
an open subset U of Rw, if JCG e Supp(jS), then there is an index H such that
&H(JCO) 7* 0; we can then choose <2i_H such that the integral
is 7^0, and such that Supp(<2!_H) is contained in an arbitrarily small neighbor-
hood V of JCG ; defining al_Rf to be 0 for H' ^ H, we obtain a form a with
support contained in V and such that f /? A a ^ 0, which proves the assertion.
In particular, for each locally integrable «-form t>, the mapping/*-* J fv is
a measure T0 on X, which is positive if and only if v(x) ^ 0 almost everywhere
(relative to the orientation of X) (13.15.3).
(17.5.1.1) Again, if /is any locally integrable complex function on X, then
the mapping DH- > \fv is an n-current T^ on X, of order 0. If U is any open
subset of X, the /2-current T<pu on X, where <PU is the characteristic function of
U, is called an open n-chain element on X, and linear combinations of open
«-chain elements are called open n-chains on X. By abuse of notation, we
shall often write U in place of TVIJ , and £ A,- Uj in place of
(17.5.2) We retain the notation and assumptions of (17.5.1). Let y be a
continuous differential #-form, where q g p; then the (n — p + <7)-form /? A 7
is locally integrable, and it follows immediately from the definitions that
(17.5.2.1) T,A, = T,Ay.