Real and Complex Analysis (Rudin)

96 REAL AND COMPLEX ANALYSIS
For instance, every Hilbert space is a Banach space, so is every I!'(p.)
normed by Ilfllp (provided we identify functions which are equal a.e.) if
1 :s; p :s; 00, and so is Co(X) with the supremum norm. The simplest Banach
space is of course the complex field itself, nonned by II x II = I x I.
One can equally well discuss real Banach spaces; the definition is exactly
the same, except that all scalars are assumed to be real.
5.3 Definition Consider a linear transformation A from a normed linear
space X into a normed linear space Y, and define its norm by
IIAII = sup·{IIAxll: x E X, Ilxll:s; 1}. (1)
If IIAII < 00, then A is called a bounded linear transformation.
In (1), Ilxll is the norm of x in X, IIAxl1 is the norm of Ax in Y; it will
frequently happen that several norms occur together, and the context will
make it clear which is which.
Observe that we could restrict ourselves to unit vectors x in (1), i.e., to x
with Ilxll = 1, without changing the supremum, since
II A(lXx) II = II IXAx II = IIX IIIAxll. (2)
Observe also that IIAII is the smallest number such that the inequality
IIAxl1 :s; IIAllllxll (3)
holds for every x E X.
The following geometric picture is helpful: A maps the closed unit ball in
X, i.e., the set
{x E X: IIxll :s; 1}, (4)
into the closed ball in Y with center at 0 and radius IIAII.
An important special case is obtained by taking the complex field for Y;
in that case we talk about bounded linear functionals.
5.4 Theorem For a linear transformation A of a normed linear space X into a
normed linear space Y, each of the following three conditions implies the other
two:
(a) A is bounded.
(b) A is continuous.
(c) A is continuous at one point of X.
PROOF Since IIA(Xl - x2)11 :s; IIAllllxl - x211, it is clear that (a) implies (b),
and (b) implies (c) trivially. Suppose A is continuous at Xo. To each E > 0 one
can then find a ~ > 0 so that Ilx - xoll < ~ implies IIAx - AXol1 < E. In other
words, IIxll < ~ implies
IIA(xo + x) - AXoll < E.