304
Appendix to Chapter III
(1.2)
Ilx + yll :::; IIxll + lIyll
as in 1R 2 or C. To see this, we define the scalar product of two vectors x and
y of CP by the formula
(1.3)
(x I y) = XlYl + ... + xPYP'
whence
Ilxll = (x I X)1/2,
and verify that the number
(zx + Y I zx + y) = (x I x)zz + (x I y)z + (x I y)z + (y I y)
is ~ 0 for any z E C; if (x I x) I- 0, we can choose
z = -(x I y)/(x I x),
and obtain the Cauchy-Schwarz inequality
(1.4)
I(x I y)1 2 :::; (x I x)(y I y) = IIxll2 ·lIy112 ,
immediately: it remains valid if (x I x) = 0 since then x = O. From this it
follows that
Ilx + yll2
(x + y I x + y) = (x I x) + (x I y) + (x I y) + (y I y) =
= IIxl12 + 2Re(x I y) + IIyl12 ::; IIxl12 + 211xll.llyll + IIyl1 2 ,
whence (2).
Metric spaces provide a much more general framework to which one can
extend almost all of what we have said about IR or C. By definition, such a
space is a pair (X, d) - one writes simply X when there is no ambiguity as
to d - formed by a set X and a function
d:Xx X ~1R+
which allows us, by convention, to define the distance of two "points"
x, y E X. We ask only for it to satisfy the obvious conditions:
(1.5)
d(x, y) ~ 0,
d(x, y) = 0 <===> x = y,
d(x, y) = d(y, x),
d(x,y):::; d(x,z) +d(z,y).
The simplest example consists of taking an arbitrary subset of a Cartesian
space for X - no matter which - and defining the distance by the formula 50 (1); more generally, every subset of a metric space can be considered as a metric space in itself. But there are also much less obvious spaces
50 One can also be less brutal. On the surface of the Earth one does not define
the distance between Paris and Heidelberg by measuring the straight line joining
these two points through the Earth. If, in a Cartesian space, one has a sufficiently
"smooth" "surface" S (of dimension possibly> 2), one measures the distance
between a, b E S by considering all the curves joining a and b on S, measuring
their lengths, and taking their greatest lower bound. In good cases there will
Axist a curve of minimum length ("geodesic"), the arc of a great circle in the
case of a sphere.
Appendix to Chapter III
(1.2)
Ilx + yll :::; IIxll + lIyll
as in 1R 2 or C. To see this, we define the scalar product of two vectors x and
y of CP by the formula
(1.3)
(x I y) = XlYl + ... + xPYP'
whence
Ilxll = (x I X)1/2,
and verify that the number
(zx + Y I zx + y) = (x I x)zz + (x I y)z + (x I y)z + (y I y)
is ~ 0 for any z E C; if (x I x) I- 0, we can choose
z = -(x I y)/(x I x),
and obtain the Cauchy-Schwarz inequality
(1.4)
I(x I y)1 2 :::; (x I x)(y I y) = IIxll2 ·lIy112 ,
immediately: it remains valid if (x I x) = 0 since then x = O. From this it
follows that
Ilx + yll2
(x + y I x + y) = (x I x) + (x I y) + (x I y) + (y I y) =
= IIxl12 + 2Re(x I y) + IIyl12 ::; IIxl12 + 211xll.llyll + IIyl1 2 ,
whence (2).
Metric spaces provide a much more general framework to which one can
extend almost all of what we have said about IR or C. By definition, such a
space is a pair (X, d) - one writes simply X when there is no ambiguity as
to d - formed by a set X and a function
d:Xx X ~1R+
which allows us, by convention, to define the distance of two "points"
x, y E X. We ask only for it to satisfy the obvious conditions:
(1.5)
d(x, y) ~ 0,
d(x, y) = 0 <===> x = y,
d(x, y) = d(y, x),
d(x,y):::; d(x,z) +d(z,y).
The simplest example consists of taking an arbitrary subset of a Cartesian
space for X - no matter which - and defining the distance by the formula 50 (1); more generally, every subset of a metric space can be considered as a metric space in itself. But there are also much less obvious spaces
50 One can also be less brutal. On the surface of the Earth one does not define
the distance between Paris and Heidelberg by measuring the straight line joining
these two points through the Earth. If, in a Cartesian space, one has a sufficiently
"smooth" "surface" S (of dimension possibly> 2), one measures the distance
between a, b E S by considering all the curves joining a and b on S, measuring
their lengths, and taking their greatest lower bound. In good cases there will
Axist a curve of minimum length ("geodesic"), the arc of a great circle in the
case of a sphere.