320
Appendix to Chapter III
of distributions) that one can establish the existence of a formula of
this type and, sometimes, write it explicitly.
The reader may never have seen a discontinuous linear map; there
is no risk of meeting them in the Cartesian spaces of finite dimension,
even less in lR. with the schoolboy functions x ..--.. ax. Be reassured: no
one has ever seen them in the case of true Banach spaces, complete 57
and of infinite dimension: it is impossible to construct one using
"natural" procedures. To obtain one requires the use of methods -
transfinite induction again - which, while affirming the existence of
such functions in the sense of logicians (and even of mathematicians
... ), never permit an effective construction because they require uncountably many arbitrary choices. The problem is the same as that
of proving the existence of a "base" for any vector space of infinite
dimension, i.e. of a family (ei)iEI of vectors such that every x E E
can be expressed in a unique way as a finite linear combination of
vectors ei; this result applies to Banach spaces and allows one to construct linear maps or forms by purely algebraic procedures; but, in
infinite dimensions, the coordinates of an x E E with respect to such
a base, though linear functions of x, are never continuous, so that
algebraic bases are of no use in analysis. One may conclude from this
that all the linear maps of one Banach space into another that can
be constructed explicitly by the standard methods of analysis, using
"formulae", are continuous. This metamathematical statement does
not, however, exempt one from proofs.
7 - Compact spaces
In the case of a subset X of C the concept of compactness has several possible
equivalent formulations (Chap. V, nO 6):
(DEF) X is closed and bounded in C;
(BW) from every sequence of points of X one can extract a subsequence which converges to a point of X;
(BL) from every covering of X by open sets one can extract a finite
covering of X.
In an arbitrary metric space X, these properties may all fail, and, also,
they are not always equivalent.
57 If one forgets this "detail", the situation changes: take C 1 (K) with the norm
IlfilK of uniform convergence and the map x ........... x'(a), where a E K is given.
To say that it is continuous means that uniform convergence of a sequence of
differentiable functions implies the convergence of their derivatives; wrong. But
CI(K) endowed with this "bad" norm is not complete.
Appendix to Chapter III
of distributions) that one can establish the existence of a formula of
this type and, sometimes, write it explicitly.
The reader may never have seen a discontinuous linear map; there
is no risk of meeting them in the Cartesian spaces of finite dimension,
even less in lR. with the schoolboy functions x ..--.. ax. Be reassured: no
one has ever seen them in the case of true Banach spaces, complete 57
and of infinite dimension: it is impossible to construct one using
"natural" procedures. To obtain one requires the use of methods -
transfinite induction again - which, while affirming the existence of
such functions in the sense of logicians (and even of mathematicians
... ), never permit an effective construction because they require uncountably many arbitrary choices. The problem is the same as that
of proving the existence of a "base" for any vector space of infinite
dimension, i.e. of a family (ei)iEI of vectors such that every x E E
can be expressed in a unique way as a finite linear combination of
vectors ei; this result applies to Banach spaces and allows one to construct linear maps or forms by purely algebraic procedures; but, in
infinite dimensions, the coordinates of an x E E with respect to such
a base, though linear functions of x, are never continuous, so that
algebraic bases are of no use in analysis. One may conclude from this
that all the linear maps of one Banach space into another that can
be constructed explicitly by the standard methods of analysis, using
"formulae", are continuous. This metamathematical statement does
not, however, exempt one from proofs.
7 - Compact spaces
In the case of a subset X of C the concept of compactness has several possible
equivalent formulations (Chap. V, nO 6):
(DEF) X is closed and bounded in C;
(BW) from every sequence of points of X one can extract a subsequence which converges to a point of X;
(BL) from every covering of X by open sets one can extract a finite
covering of X.
In an arbitrary metric space X, these properties may all fail, and, also,
they are not always equivalent.
57 If one forgets this "detail", the situation changes: take C 1 (K) with the norm
IlfilK of uniform convergence and the map x ........... x'(a), where a E K is given.
To say that it is continuous means that uniform convergence of a sequence of
differentiable functions implies the convergence of their derivatives; wrong. But
CI(K) endowed with this "bad" norm is not complete.