Appendix to Chapter III
319
(iii) (for spaces over JR.) Every closed convex subset C of E can be
defined by a family of inequalities fi (x) 2:: ai where the Ii are clf
and the ai are real constants (in other words, C is an intersection
of "closed half spaces").
One cannot prove these theorems in full generality without using the
methods of "transfinite induction" in one place or another; these are
founded on the ordinals and the axiom of choice of Chap. I, nO 9; they
were not presented because totally unnecessary at the level of this
treatise. Nevertheless these theorems lead to very important concrete
results in analysis when applied to functional spaces. In this case, the
existence of a not identically zero clf is always obvious in practice,
but properties (i), (ii) and (iii) are not.
Another important result. Consider two Banach spaces E and F
and endow their Cartesian product E x F with the obvious vector
structure and with the norm II(x, y)1I = Ilxll + lIyll. One obtains a new
Banach space. Consider now a linear map h : E ~ F; the graph of
h, i.e. the set H c E x F of pairs (x, h(x)), is then clearly a vector
subspace of E x F. The closed graph theorem (Banach-Steinhaus, two
interwar Poles) says that, for h to be continuous, it is necessary and
sufficient that its graph be a closed vector subspace of E x F. The
necessity of the condition is immediate, but the converse requires ingenious arguments which we do not reproduce, and which use Baire's
Theorem mentioned in Chap. III, end of nO 6, note -12.
Let us explain the interest of the theorem by considering a linear
map 9 : E ~ F which is both continuous and bijective. It must
have an inverse map h : F ~ E, such that
x = h(y) {=} y = g(x).
It is clear that h is linear like 9 and that the graph H of h is the
image of the graph G of 9 under the "symmetry" (x, y) f--+ (y, x)
(Chap. I, n° 5). This clearly transforms the closed subsets of Ex F
into closed subsets of F x E since it preserves distances.
Since 9 is continuous, it is clear that G is closed; so similarly is H.
Conclusion: a bijective continuous linear map of one Banach space
onto another is "bicontinuous", i.e. possesses a continuous inverse
map, i.e. is an homeomorphism. A reassuring result, but it depends
on the closed graph theorem.
The example of the map x f--+ x' of Cl(K) into CO(K) does
not bring out the power of the closed graph theorem, since, in this
case, one can verify it directly by more elementary methods. But it
is otherwise when one considers functions of several variables and
maps of the "linear differential operator" kind, for which one does
not a priori have a formula as simple as (5); it is on the contrary,
thanks to the closed graph theorem (and, generally, to the theory
319
(iii) (for spaces over JR.) Every closed convex subset C of E can be
defined by a family of inequalities fi (x) 2:: ai where the Ii are clf
and the ai are real constants (in other words, C is an intersection
of "closed half spaces").
One cannot prove these theorems in full generality without using the
methods of "transfinite induction" in one place or another; these are
founded on the ordinals and the axiom of choice of Chap. I, nO 9; they
were not presented because totally unnecessary at the level of this
treatise. Nevertheless these theorems lead to very important concrete
results in analysis when applied to functional spaces. In this case, the
existence of a not identically zero clf is always obvious in practice,
but properties (i), (ii) and (iii) are not.
Another important result. Consider two Banach spaces E and F
and endow their Cartesian product E x F with the obvious vector
structure and with the norm II(x, y)1I = Ilxll + lIyll. One obtains a new
Banach space. Consider now a linear map h : E ~ F; the graph of
h, i.e. the set H c E x F of pairs (x, h(x)), is then clearly a vector
subspace of E x F. The closed graph theorem (Banach-Steinhaus, two
interwar Poles) says that, for h to be continuous, it is necessary and
sufficient that its graph be a closed vector subspace of E x F. The
necessity of the condition is immediate, but the converse requires ingenious arguments which we do not reproduce, and which use Baire's
Theorem mentioned in Chap. III, end of nO 6, note -12.
Let us explain the interest of the theorem by considering a linear
map 9 : E ~ F which is both continuous and bijective. It must
have an inverse map h : F ~ E, such that
x = h(y) {=} y = g(x).
It is clear that h is linear like 9 and that the graph H of h is the
image of the graph G of 9 under the "symmetry" (x, y) f--+ (y, x)
(Chap. I, n° 5). This clearly transforms the closed subsets of Ex F
into closed subsets of F x E since it preserves distances.
Since 9 is continuous, it is clear that G is closed; so similarly is H.
Conclusion: a bijective continuous linear map of one Banach space
onto another is "bicontinuous", i.e. possesses a continuous inverse
map, i.e. is an homeomorphism. A reassuring result, but it depends
on the closed graph theorem.
The example of the map x f--+ x' of Cl(K) into CO(K) does
not bring out the power of the closed graph theorem, since, in this
case, one can verify it directly by more elementary methods. But it
is otherwise when one considers functions of several variables and
maps of the "linear differential operator" kind, for which one does
not a priori have a formula as simple as (5); it is on the contrary,
thanks to the closed graph theorem (and, generally, to the theory