§2. Uniform convergence
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the function x on E = [0, 1[ has least upper bound 1, but x < 1 for all x E E.
Nevertheless, there always exists a sequence (xn ) of points of E such that
(7.2)
lim/(xn ) = sup/(x);
it suffices to choose Xn so that I(xn ) > u - lin, as allowed by (SUP 2').
There are analogous definitions and notation for the case of greatest lower
bounds. When the least upper bound is infinite, the condition (SUP 1) is
empty and (SUP 2') means that for all A E IR there is a point x in X where
I(x) > A.
If two maps I, 9 : E ----+ IR are bounded above or below then clearly so
is the function I + 9 and, with or without these hypotheses, we have the
inequalities
(7.3')
(7.3")
sup(f(x) + g(x)) < sup I(x) + sup g(x),
inf(f(x) + g(x)) > inf I(x) + inf g(x).
If one puts sup/(x) = A and supg(x) = B, then I(x) ::; A and g(x) ::; B for
all x, whence I(x) + g(x) ::; A + B, and so (3').
If, further, the values of I and 9 are positive and bounded above then the
map I 9 is also bounded above, since one can multiply inequalities between
positive numbers member-by-member.
These definitions are not meaningful for functions with complex values.
In this case one can define the positive number
(7.4)
lilliE = sup I/(x)1 ::; +00,
xEE
the unilorm norm 01 I on E for every set E on which I is defined. If I is
bounded on E, Le. if there exist numbers M ::.:: 0 such that I/(x)1 ::; M for
all x E E or, this comes to the same, if lilliE < +00, this norm is then the
least possible M. One can thus always, and generally advantageously, replace
a bound of the form
I/(x)1 ::; M for all x E E
by lilliE ::; M: the two relations are strictly equivalent and the second is
more concise.
It is clear that if I and g, with complex values, are bounded, so similarly
are 1+ 9 and Ig; then
(7.5)
(7.6)
III + gilE < lilliE + IIgIIE'
IIlgllE < 1I/IIE·lIgIIE
since the right hand sides majorise I/(x) + g(x)1 and I/(x)g(x)1 for any x. It
is even more obvious that