316
Appendix to Chapter III
but this limit is < r for p large by definition of a Cauchy sequence. So
lim IIf - fpI12 = 0, qed.
One can show that this procedure yields all Hilbert spaces 54 .
Example 2. Let X be an interval in JR and consider the complex vector space
L(X) of continuous functions of compact support on X (Chap. V, nO 31; if
X is compact, this is the set of all continuous functions on X). For two such
functions, put
(5.7)
(f 1 g) = / f(x)g(x)dx
where one integrates over X. The axioms (H 1), ... , (H 4) are again satisfied,
(H 4) because one restricts to continuous functions. The corresponding norm
is
IIfl12 = (/ If(x)1 2 dX) 1/2
You may clearly replace dx by any positive Radon measure dJ.L(x) on X, and
X by any locally compact subset of C (Chap. III, nO 31). One thus obtains a
non complete pre-Hilbert space. Lebesgue integration allows us to construct
true Hilbert spaces by this type of procedure.
These two types of space play a large role in the theory of Fourier series
and, in fact, are the origin of the concept of pre-Hilbert space; it needed
fully a quarter of a century to pass from these "concrete" pre-Hilbert spaces,
whose elements are functions, to the "abstract" Hilbert spaces whose theory
(J. von Neumann), much richer than that of Banach spaces, dates from the
end of the 1920s and later became the subject of much research.
Analysis provides an inexhaustible stock of examples of Banach spaces.
The most obvious is the set of bounded scalar functions on a set X, endowed
with the norm IIfllx of uniform convergence. A less trivial example is the
vector space CP(K) of functions of class CP, p finite, on a compact interval
K c JR, endowed with the norm
(5.8)
a finite expression, since, on a compact set, a continuous function is necessarily bounded (Chap. III, n° 9, Theorem 11). As we showed in n° 3, with
the help of Chap. III, nO 17, Theorem 20, this space is complete.
6 - Continuous linear maps
In practice, almost all the theorems about infinite dimensional Banach spaces
concern continuous linear maps55 f : E ---+ F from one Banach space into
54 More precisely: if 1t is an Hilbert space there exists a set X and a linear bijective
map of 1t onto L2(X) which preserves the scalar products (an "isomorphism" of
Hilbert spaces).
55 There are exceptions, for example the fixed point theorem for the equation
f(x) = x when f satisfies a condition Jlf(x') - f(x")JI ~ qJlx' - x"Jl with q < 1.
Appendix to Chapter III
but this limit is < r for p large by definition of a Cauchy sequence. So
lim IIf - fpI12 = 0, qed.
One can show that this procedure yields all Hilbert spaces 54 .
Example 2. Let X be an interval in JR and consider the complex vector space
L(X) of continuous functions of compact support on X (Chap. V, nO 31; if
X is compact, this is the set of all continuous functions on X). For two such
functions, put
(5.7)
(f 1 g) = / f(x)g(x)dx
where one integrates over X. The axioms (H 1), ... , (H 4) are again satisfied,
(H 4) because one restricts to continuous functions. The corresponding norm
is
IIfl12 = (/ If(x)1 2 dX) 1/2
You may clearly replace dx by any positive Radon measure dJ.L(x) on X, and
X by any locally compact subset of C (Chap. III, nO 31). One thus obtains a
non complete pre-Hilbert space. Lebesgue integration allows us to construct
true Hilbert spaces by this type of procedure.
These two types of space play a large role in the theory of Fourier series
and, in fact, are the origin of the concept of pre-Hilbert space; it needed
fully a quarter of a century to pass from these "concrete" pre-Hilbert spaces,
whose elements are functions, to the "abstract" Hilbert spaces whose theory
(J. von Neumann), much richer than that of Banach spaces, dates from the
end of the 1920s and later became the subject of much research.
Analysis provides an inexhaustible stock of examples of Banach spaces.
The most obvious is the set of bounded scalar functions on a set X, endowed
with the norm IIfllx of uniform convergence. A less trivial example is the
vector space CP(K) of functions of class CP, p finite, on a compact interval
K c JR, endowed with the norm
(5.8)
a finite expression, since, on a compact set, a continuous function is necessarily bounded (Chap. III, n° 9, Theorem 11). As we showed in n° 3, with
the help of Chap. III, nO 17, Theorem 20, this space is complete.
6 - Continuous linear maps
In practice, almost all the theorems about infinite dimensional Banach spaces
concern continuous linear maps55 f : E ---+ F from one Banach space into
54 More precisely: if 1t is an Hilbert space there exists a set X and a linear bijective
map of 1t onto L2(X) which preserves the scalar products (an "isomorphism" of
Hilbert spaces).
55 There are exceptions, for example the fixed point theorem for the equation
f(x) = x when f satisfies a condition Jlf(x') - f(x")JI ~ qJlx' - x"Jl with q < 1.