30 NONLINEAR REPRESENTATION AND SPACES
for some constant C. It follows from inequalities (2.19) and (2.31) that (with a new
constant C)
qi(u) C\\u\\Ei, ueE^.
By continuity this inequality is true for wG^i , which proves the lemma.
To state the next theorem we introduce the following seminorms qn, n 0, on E^:
(qn(u))2 = (q™(v))2 + (?f (a))2, u = (v,a) E E^v e M ^ Q G LOO, (2.32a)
/Lii/n
and
qZ(a) =
C£\\M»d"a\\2D)\
(2.32b)
/xn
\u\n
where /i and v are multiindices and
MM
is defined as in Lemma 2.8. As will be proved, qn
has a continuous extension to En.
Theorem 2.9. There exist constants Cn 0 such that
c^lMk ^feM
^cnh\\En,
no.
Moreover the linear space
EC = MC®DC = {(/,/, a) € EoolfJ €
(^(IR3

{0},C4),a
€
C0°°(K3,C4)}
25 dense in E^.
Proof. According to definition (1.6a) of  • \\E and expression (2.12) for Tyix, Y G II',
u — (/, / , a) € £oo of Lemma 2.7 we have
Hlin = £
(\\v(Y)*\\l*
+ WWiQMf + ^(r)/)!!^ (2.33)
Yen'
\Y\n
+
vr1(i?2(r)/
+
Q2(F)/)22).
We shall estimate the terms on the righthand side of (2.33). It follows at once from
inequality (2.14) in Lemma 2.7 that
r(F)oL2 Cntf(a), YeW,\Y\n, (2.34)
and from (2.13a) that
(IIM'QaPO/lli* + llVr 1 Q
2
(y)/l
2
)* Cnq™(fJ), Y € n',F  n. (2.35)