242 XVII DISTRIBUTIONS AND DIFFERENTIAL OPERATORS
#P(X) of a linear form T' on ^r)(X) whose restriction to each ^r)(X; K) is
continuous, then T is of order ^r. Conversely, we shall show that if T is of
order ^r, then it is the restriction of such a linear form T', which moreover is
unique. Let K be a compact subset of X and let K' be a compact neighborhood
of K in X. Then there exists a C^-function h which is equal to 1 on a compact
neighborhood of K and which is 0 on X - K' (16.4.2). For each p-form
f$ e S>%\X; K), there exists a sequence &k in @P(X) which converges to j8 with
respect to the topology of ^r)(X) (17.2.2). The sequence (/zak), which belongs
to @P(X; K'), therefore also converges to p in ^r)(X; K') by (17.1.4). In
other words, the closure of @P(X; K') in ^(X) contains ^r)(X; K) and is
contained in ^r)(X; K'). The existence and uniqueness of T' now follow
immediately by applying (12.9.4). We shall often write T in place of T'.
Examples of currents
(17.3.3) Let jc be a point of X, and let z^ be a tangent p-vector at x (i.e., an
element of /\ T^X)). Then the mapping an^<zx, a(X)> is a continuous linear
form on <f£0)(X), hence is a p-current of order 0; it is called the Dirac p-current
defined by xx9 and is sometimes denoted by ezx. When p = 0, we obtain the
scalar multiples of the Dirac measure (13.1.3).
(17.3.4) It follows from (17.3.2) that distributions of order 0 on X are con-
tinuous linear forms on each of the Banach spaces @$\X; K) = ^T(X; K),
and are therefore precisely the (complex) measures on X.
(17.3.5) Let T be a p-current and co a C°° differential #-form (i.e., an element
of £q(X))9 with q^p. For each (p-0)-form /?e®p.,(X; K), we have
co A£ e @P(X; K), and it follows therefore from (17.2.3) that the linear form
/?i-»T(co A /?) is a (p - #)-current, which is denoted by T A co. If T is of order
^r, then so is TAG), and in this case we can also define T A CD when co is a
differential #-form of class Cr. When q — 0, so that co is a complex-valued
function g, we write T • g or g • T in place of T A co; if T is a measure, this
definition agrees with that of (13.1.5), because of the fact that the closure of
0(X; K') in #C(X) contains JT(X; K) for each compact neighborhood K'
ofK.
(17.3.6) Suppose that p > 0, and let Y be a C°° vector field on X. For each
p-form a e &p(X; K), we have IY • a e %-x(X; K) (16.18.4). For each (p - 1)-
current T, it therefore follows again from (17.2.3) that the linear form
aj->T(/r • a) is a p-current, which is of order ^r if T is of order ^r, and which
is denoted by \ - T.