318
Appendix to Chapter III
and x' depend; here is an example of a case where the "functions" become
"variables" ]. It is elear that f is linear, and continuous since Ilf(x)11 ::::
Ilx'II K ::; IIxll, by (3).
Is the map f bijective? It would first have to be injective, Le. f(x) :::: 0
must imply x = 0; but f(x) = 0 means that the function x(t) has an identically zero derivative; so it is constant, but not necessarily zero. One must
therefore eliminate the constants from E, which can be done by imposing
the supplementary condition x(a) = 0 on the functions of E, for an a E K
chosen once and for all. Having so modified E, f becomes injective. It is then
surjective since a continuous function y(t) on K always admits (Chap. V,
nO 12) a primitive x(t) such that x'(t) = y(t), x(a) = o.
We can therefore consider the inverse map 9 : F ----+ E of f. It associates
to each y E F the primitive x = g(y) E E which vanishes at t = a, namely
(6.5)
x(t) = it y(u)du.
The map 9 is continuous because both
IlxliK ::; m(K)IIYIIK and IIx'IIK = IlyllK,
whence IIxll ~ (1 + m(K») Ilyli.
There are important theorems in the theory of Banach spaces.
The first is the Hahn-Banach theorem, which, at the lowest level,
assures the existence, in any Banach space E, of nontrivial continuous
linear forms (elf), i.e. of linear maps f : E ----+ IR or C which are
continuous and not identically zero; example: E = CO(K), where K
is a compact subset of IR or C; the elf are exactly, by definition, the
Radon measures on K (Chap. V, nO 30). There are several variants
of the theorem.
(i) For every nonzero a E E there exists a elf f such that f(a) =I=- o.
More generally: if F is a elosed vector subspace of E and if 9 is
a elf on F, then there exists a elf on E such that f = 9 in F.
(ii) Every closed vector subspace F of E can be defined by a family
(in general infinite) of equations fi(X) = 0 where the fi are elp6
on E.
56 For example take E = CD(K), where K is a compact interval in JR, and for F the
set of functions x E E which are uniform limits on K of polynomials; it is a closed
vector subspace (closure of the set of polynomials) of E. Since the clf on E are
the (complex) Radon measures on K, the Weierstrass approximation theoremwhich affirms that F = E - is thus equivalent to the following assertion: the only
measure /.t on K such that /.t(p) = 0 for all polynomials, i.e. J tnd/.t(t) = 0 for all
n E N, is /.t = o.
Appendix to Chapter III
and x' depend; here is an example of a case where the "functions" become
"variables" ]. It is elear that f is linear, and continuous since Ilf(x)11 ::::
Ilx'II K ::; IIxll, by (3).
Is the map f bijective? It would first have to be injective, Le. f(x) :::: 0
must imply x = 0; but f(x) = 0 means that the function x(t) has an identically zero derivative; so it is constant, but not necessarily zero. One must
therefore eliminate the constants from E, which can be done by imposing
the supplementary condition x(a) = 0 on the functions of E, for an a E K
chosen once and for all. Having so modified E, f becomes injective. It is then
surjective since a continuous function y(t) on K always admits (Chap. V,
nO 12) a primitive x(t) such that x'(t) = y(t), x(a) = o.
We can therefore consider the inverse map 9 : F ----+ E of f. It associates
to each y E F the primitive x = g(y) E E which vanishes at t = a, namely
(6.5)
x(t) = it y(u)du.
The map 9 is continuous because both
IlxliK ::; m(K)IIYIIK and IIx'IIK = IlyllK,
whence IIxll ~ (1 + m(K») Ilyli.
There are important theorems in the theory of Banach spaces.
The first is the Hahn-Banach theorem, which, at the lowest level,
assures the existence, in any Banach space E, of nontrivial continuous
linear forms (elf), i.e. of linear maps f : E ----+ IR or C which are
continuous and not identically zero; example: E = CO(K), where K
is a compact subset of IR or C; the elf are exactly, by definition, the
Radon measures on K (Chap. V, nO 30). There are several variants
of the theorem.
(i) For every nonzero a E E there exists a elf f such that f(a) =I=- o.
More generally: if F is a elosed vector subspace of E and if 9 is
a elf on F, then there exists a elf on E such that f = 9 in F.
(ii) Every closed vector subspace F of E can be defined by a family
(in general infinite) of equations fi(X) = 0 where the fi are elp6
on E.
56 For example take E = CD(K), where K is a compact interval in JR, and for F the
set of functions x E E which are uniform limits on K of polynomials; it is a closed
vector subspace (closure of the set of polynomials) of E. Since the clf on E are
the (complex) Radon measures on K, the Weierstrass approximation theoremwhich affirms that F = E - is thus equivalent to the following assertion: the only
measure /.t on K such that /.t(p) = 0 for all polynomials, i.e. J tnd/.t(t) = 0 for all
n E N, is /.t = o.