314
Appendix to Chapter III
the XX th century53, considerably advanced Logic, Set Theory, Topology
Functional Analysis, etc. The foremost theorem - this is a lot to say .. :
- is that, in a Banach space, every absolutely convergent series converges.
The calculations of nO 1 which led us to the Cauchy-Schwarz inequality
are purely formal; the fact that they concern ''vectors'' in lR P or CP (rather
than integrable functions, for example, as we shall see in Chap. V, n° 5 or in
Chap. VII, nO 7) is immaterial; only the algebraic rules of calculating with
scalar products are important here. So we are led to a vast generalisation ,
and to define a pre-Hilbert space as follows: this is a vector space 1t over
C, in general of infinite dimension, in which one is given a "scalar product"
satisfying the obvious conditions:
(H 1): (x I y) is a linear function of x for y given;
(H 2): (y I x) = (x I y);
(H 3): (x I x) :2: 0 for all x.
The space is said to be separated if it satisfies the condition
(H 4): (x I x) = 0 implies x = o.
Again we have the Cauchy-Schwarz inequality, a norm Ilxll = (x I x)1/2 which
can vanish for x -I 0 if 1t is not separated, and a triangle inequality that one
writes in the form (1), or as d(x, y) ::; d(x, z) + d(z, y), where one defines
d(x,y) = Ilx - yll = (x - y I x - y)t. It might happen that the relation
d(x, y) = 0 no longer implies x = y: axiom (H 4) serves to ensure this. A
separated pre-Hilbert space is thus a normed vector space; if it is complete,
one says that it is a Hilbert space.
In contrast, if K C lR is a compact interval, the space GO(K) endowed
with the distance
induced by the scalar product
(f I g) = L f(x)g(x)dx
or, even more simply, by the distance d1(f,g) = J If(x) - g(x)ldx, is not
complete; similarly if one considers the set of all Riemann integrable functions
on K. On the other hand, the Lebesgue theory leads to complete spaces.
53 The situation changed with the war: the Jews emigrated or were exterminated
and the "Aryans" who survived after 1945 suffered materially very difficult conditions for a long time, and had hardly any contacts except with the Soviet
mathematicians (and they too ... ), not nothing, but not replacing the rest of
the world. Similar situation in Hungary. In their period of greatness the mathematicians of these two "small" countries were, together, much more "modern"
than the majority of their French, English, American, or even German, counterparts.
Appendix to Chapter III
the XX th century53, considerably advanced Logic, Set Theory, Topology
Functional Analysis, etc. The foremost theorem - this is a lot to say .. :
- is that, in a Banach space, every absolutely convergent series converges.
The calculations of nO 1 which led us to the Cauchy-Schwarz inequality
are purely formal; the fact that they concern ''vectors'' in lR P or CP (rather
than integrable functions, for example, as we shall see in Chap. V, n° 5 or in
Chap. VII, nO 7) is immaterial; only the algebraic rules of calculating with
scalar products are important here. So we are led to a vast generalisation ,
and to define a pre-Hilbert space as follows: this is a vector space 1t over
C, in general of infinite dimension, in which one is given a "scalar product"
satisfying the obvious conditions:
(H 1): (x I y) is a linear function of x for y given;
(H 2): (y I x) = (x I y);
(H 3): (x I x) :2: 0 for all x.
The space is said to be separated if it satisfies the condition
(H 4): (x I x) = 0 implies x = o.
Again we have the Cauchy-Schwarz inequality, a norm Ilxll = (x I x)1/2 which
can vanish for x -I 0 if 1t is not separated, and a triangle inequality that one
writes in the form (1), or as d(x, y) ::; d(x, z) + d(z, y), where one defines
d(x,y) = Ilx - yll = (x - y I x - y)t. It might happen that the relation
d(x, y) = 0 no longer implies x = y: axiom (H 4) serves to ensure this. A
separated pre-Hilbert space is thus a normed vector space; if it is complete,
one says that it is a Hilbert space.
In contrast, if K C lR is a compact interval, the space GO(K) endowed
with the distance
induced by the scalar product
(f I g) = L f(x)g(x)dx
or, even more simply, by the distance d1(f,g) = J If(x) - g(x)ldx, is not
complete; similarly if one considers the set of all Riemann integrable functions
on K. On the other hand, the Lebesgue theory leads to complete spaces.
53 The situation changed with the war: the Jews emigrated or were exterminated
and the "Aryans" who survived after 1945 suffered materially very difficult conditions for a long time, and had hardly any contacts except with the Soviet
mathematicians (and they too ... ), not nothing, but not replacing the rest of
the world. Similar situation in Hungary. In their period of greatness the mathematicians of these two "small" countries were, together, much more "modern"
than the majority of their French, English, American, or even German, counterparts.