218
III - Convergence: Continuous variables
Ilc·fllE = 1el-llfllE
for every constant c E C. The associativity formula for least upper bounds
established at the end of nO 9 of Chap. II can be translated as follows: if f is
a scalar function defined on the union E = U Ei , i E I, of any family of sets,
then
IlfilE = sup IlfllEi •
iEI
When f and 9 are functions with complex values defined on E, we define
their uniform distance on E by
(7.7)
dE(f, g) = Ilf - gllE = sup If(x) - g(x)1 = supd[f(x), g(x)].
The left hand side of (7), if it is finite (for example if f and 9 are bounded,
not a necessary condition), is thus the least number M ~ 0 such that
(7.8)
If(x) - g(x)1 :S M for any x E X.
The inequality (5) shows that, if f, 9 and h are three bounded functions
on E then
(7.9)
dE(f,g) :S dE(f, h) + dE(h,g).
It is moreover clear that dE(f,g) ~ 0 always, and that this distance can be
zero only if f = g: for then 0 majorises all the numbers If(x) - g(x)l.
The uniform convergence on E of a sequence of functions (fn) to a limit
function f can be translated immediately into these terms. One clearly does
not change the situation if one replaces strict by weak inequality in (5.5).
But to say, for a given n, that d[f(x), fn(x)] :S r for any x E E is the same
as saying that dE(f, fn) :S r. Since, for all r > 0, this relation must be
satisfied for n sufficiently large, one concludes that uniform convergence on
E is equivalent to the relation
(7.10)
analogously to the convergence of a sequence of numbers. This statement,
trivial though it is, will be of constant use.
From it one deduces easily, as in the case of scalar sequences, the rules
of algebraic calculations under uniform convergence; using them often allows
us to bypass explicit calculations.
(R 1) If two sequences (fn) and (gn) converge uniformly to f and 9 then
the sequence (fn + gn) converges uniformly to f + g.
III - Convergence: Continuous variables
Ilc·fllE = 1el-llfllE
for every constant c E C. The associativity formula for least upper bounds
established at the end of nO 9 of Chap. II can be translated as follows: if f is
a scalar function defined on the union E = U Ei , i E I, of any family of sets,
then
IlfilE = sup IlfllEi •
iEI
When f and 9 are functions with complex values defined on E, we define
their uniform distance on E by
(7.7)
dE(f, g) = Ilf - gllE = sup If(x) - g(x)1 = supd[f(x), g(x)].
The left hand side of (7), if it is finite (for example if f and 9 are bounded,
not a necessary condition), is thus the least number M ~ 0 such that
(7.8)
If(x) - g(x)1 :S M for any x E X.
The inequality (5) shows that, if f, 9 and h are three bounded functions
on E then
(7.9)
dE(f,g) :S dE(f, h) + dE(h,g).
It is moreover clear that dE(f,g) ~ 0 always, and that this distance can be
zero only if f = g: for then 0 majorises all the numbers If(x) - g(x)l.
The uniform convergence on E of a sequence of functions (fn) to a limit
function f can be translated immediately into these terms. One clearly does
not change the situation if one replaces strict by weak inequality in (5.5).
But to say, for a given n, that d[f(x), fn(x)] :S r for any x E E is the same
as saying that dE(f, fn) :S r. Since, for all r > 0, this relation must be
satisfied for n sufficiently large, one concludes that uniform convergence on
E is equivalent to the relation
(7.10)
analogously to the convergence of a sequence of numbers. This statement,
trivial though it is, will be of constant use.
From it one deduces easily, as in the case of scalar sequences, the rules
of algebraic calculations under uniform convergence; using them often allows
us to bypass explicit calculations.
(R 1) If two sequences (fn) and (gn) converge uniformly to f and 9 then
the sequence (fn + gn) converges uniformly to f + g.