322
Appendix to Chapter III
a Banach space X, though closed and bounded in X, is not compact except
in finite dimensions, a famous theorem of F. Riesz (Dieudonne, Vol. 1, V.g).
The image of a compact space under a continuous map is a compact set
and the inverse map, if it exists, is continuous. On a compact space, every
continuous function (with values in a metric space) is uniformly continuous
and bounded; if it is real, it attains its maximum and its minimum. In a
compact space, uniform convergence is a property of a local nature. The
product X x Y of two compact spaces X and Y is compact [use (BW)]. In
a compact space, any closed set is compact and conversely; the union of a
finite number of such sets is compact [use (BW)]. Etc.
One can also, more generally, define locally compact spaces by the condition that every point possesses a compact neighbourhood.
8 - Topological spaces
We can generalise further by introducing topological spaces, the final and
definitive culmination of all these theories. This is an object formed by a
set X and a set of subsets of X, conventionally called "open", among which
must be the set X and the empty set, and required only to satisfy the two
standard properties of open sets in R or C: every union of open sets is open;
every finite intersection of open sets is open.
The concept of continuity is introduced as follows. Given two topological
spaces X and Y and a map f : X --+ Y, one says that f is continuous at
a E X if, for any open V in Y containing b = f(a), there exists an open U
in X containing a such that
x E U ===} f(x) E V.
Further, a sequence of points Xn of X converges to a limit a E X if, for
every open set U containing a, one has Xn E U for all sufficiently large n. A
point a will be said to be interior to a set E c X if E contains an open set
containing a. Etc.
A quite general method for defining a topology on a set X consists of
specifying a family (di)iEI of maps of X x X into R+, each having the properties of a distance with the exception of the relation di(x, y) = 0 ===} x = y.
The role of open balls with centre a E X is then taken by the sets defined by
a finite number of inequalities di(a, x) < Ti, the open sets then being defined
as in metric spaces.
For example, take X = GO(R) and, for all k E 1' \:1, put
(8.1)
where
dk(f, g) = sup If(x) - g(x)1 = vk(f - g)
Ixl~k
Appendix to Chapter III
a Banach space X, though closed and bounded in X, is not compact except
in finite dimensions, a famous theorem of F. Riesz (Dieudonne, Vol. 1, V.g).
The image of a compact space under a continuous map is a compact set
and the inverse map, if it exists, is continuous. On a compact space, every
continuous function (with values in a metric space) is uniformly continuous
and bounded; if it is real, it attains its maximum and its minimum. In a
compact space, uniform convergence is a property of a local nature. The
product X x Y of two compact spaces X and Y is compact [use (BW)]. In
a compact space, any closed set is compact and conversely; the union of a
finite number of such sets is compact [use (BW)]. Etc.
One can also, more generally, define locally compact spaces by the condition that every point possesses a compact neighbourhood.
8 - Topological spaces
We can generalise further by introducing topological spaces, the final and
definitive culmination of all these theories. This is an object formed by a
set X and a set of subsets of X, conventionally called "open", among which
must be the set X and the empty set, and required only to satisfy the two
standard properties of open sets in R or C: every union of open sets is open;
every finite intersection of open sets is open.
The concept of continuity is introduced as follows. Given two topological
spaces X and Y and a map f : X --+ Y, one says that f is continuous at
a E X if, for any open V in Y containing b = f(a), there exists an open U
in X containing a such that
x E U ===} f(x) E V.
Further, a sequence of points Xn of X converges to a limit a E X if, for
every open set U containing a, one has Xn E U for all sufficiently large n. A
point a will be said to be interior to a set E c X if E contains an open set
containing a. Etc.
A quite general method for defining a topology on a set X consists of
specifying a family (di)iEI of maps of X x X into R+, each having the properties of a distance with the exception of the relation di(x, y) = 0 ===} x = y.
The role of open balls with centre a E X is then taken by the sets defined by
a finite number of inequalities di(a, x) < Ti, the open sets then being defined
as in metric spaces.
For example, take X = GO(R) and, for all k E 1' \:1, put
(8.1)
where
dk(f, g) = sup If(x) - g(x)1 = vk(f - g)
Ixl~k