Appendix to Chapter III
317
another. The continuity of such a map translates into a simple inequality.
Since f(O) = 0, there is a number e > 0 such that Ilxll $ e implies IIf(x)II $ 1.
Now, for all x E E, one has IiAxil $ e if one chooses A = e/llxll; since
f(AX) = IAlf(x) = e·f(x)/IIxll,
one has ellf(x)II/IIxil $ 1 for all x, in other words
(6.1)
IIf(x)II $ M·llxll
where M = lie is a constant> 0 independent of x. Conversely, (1) implies
d[f(x), fey)] = IIf(x) - f(y)II = IIf(x - y)II $ M·llx - yll
for all x, y E E, whence continuity. The latter, for linear maps, is thus equivalent to the existence of an inequality of the form (1). The least possible
constant M is called the norm of f, written Ilfll.
If f is bijective it possesses an inverse map g, clearly linear like f. If g is
continuous there is a relation of the form IIg(y)II $ M'IIyll, with a constant
M' > O. Since every y E Y is of the form f(x) with x = g(y), the relation in
question can also be written as IIxil $ M'.IIf(x)II. So we conclude that if a
bijective continuous linear map f : E ~ F has a continuous inverse g then
(6.2)
m·llxll $ IIf(x)II $ M·llxll
with two strictly positive constants m = 11M' and M.
Exercise - A linear map satisfying (2) is injective, maps E onto a closed
vector subspace H = f(E) of F and the inverse map H ~ E is continuous.
Note in passing that a closed vector subspace of a Banach space is itself a
Banach space (and, more generally, that every closed set in a complete metric
space is itself a complete metric space).
Let us give a simple example of a continuous linear map using Chap. V.
Consider, on a compact interval K c JR., the spaces E = C1(K) and
F = COCK), E being endowed with the norm
(6.3)
Ilxil = sup Ix(t)1 + sup Ix'(t)1 = IIxliK + IIx'ilK
tEK
tEK
and F with the norm
(6.4)
IIyll = sup ly(t)1 = IIyllK.
tEK
E and F become Banach spaces in this way, since they are complete, as we
have seen above. Now consider the map f : E ~ Y, which, to each function
x E E, associates its derivative f(x) = x' [it is not without reason that,
contrary to our habits, we have denoted by t the real variable on which x
317
another. The continuity of such a map translates into a simple inequality.
Since f(O) = 0, there is a number e > 0 such that Ilxll $ e implies IIf(x)II $ 1.
Now, for all x E E, one has IiAxil $ e if one chooses A = e/llxll; since
f(AX) = IAlf(x) = e·f(x)/IIxll,
one has ellf(x)II/IIxil $ 1 for all x, in other words
(6.1)
IIf(x)II $ M·llxll
where M = lie is a constant> 0 independent of x. Conversely, (1) implies
d[f(x), fey)] = IIf(x) - f(y)II = IIf(x - y)II $ M·llx - yll
for all x, y E E, whence continuity. The latter, for linear maps, is thus equivalent to the existence of an inequality of the form (1). The least possible
constant M is called the norm of f, written Ilfll.
If f is bijective it possesses an inverse map g, clearly linear like f. If g is
continuous there is a relation of the form IIg(y)II $ M'IIyll, with a constant
M' > O. Since every y E Y is of the form f(x) with x = g(y), the relation in
question can also be written as IIxil $ M'.IIf(x)II. So we conclude that if a
bijective continuous linear map f : E ~ F has a continuous inverse g then
(6.2)
m·llxll $ IIf(x)II $ M·llxll
with two strictly positive constants m = 11M' and M.
Exercise - A linear map satisfying (2) is injective, maps E onto a closed
vector subspace H = f(E) of F and the inverse map H ~ E is continuous.
Note in passing that a closed vector subspace of a Banach space is itself a
Banach space (and, more generally, that every closed set in a complete metric
space is itself a complete metric space).
Let us give a simple example of a continuous linear map using Chap. V.
Consider, on a compact interval K c JR., the spaces E = C1(K) and
F = COCK), E being endowed with the norm
(6.3)
Ilxil = sup Ix(t)1 + sup Ix'(t)1 = IIxliK + IIx'ilK
tEK
tEK
and F with the norm
(6.4)
IIyll = sup ly(t)1 = IIyllK.
tEK
E and F become Banach spaces in this way, since they are complete, as we
have seen above. Now consider the map f : E ~ Y, which, to each function
x E E, associates its derivative f(x) = x' [it is not without reason that,
contrary to our habits, we have denoted by t the real variable on which x