Real and Complex Analysis (Rudin)

CHAPTER
EIGHTEEN
ELEMENTARY THEORY OF
BANACH ALGEBRAS
Introduction
356
18.1 Definitions A complex algebra is a vector space A over the complex field
in which an associative and distributive multiplication is defined, i.e.,
x(yz) = (xy)z, (x + y)z = xz + yz, x(y + z) = xy + xz (1)
for x, y, and z E A, and which is related to scalar multiplication so that
ex(xy) = x(exy) = (exx)y (2)
for x and YEA, ex a scalar.
If there is a norm defined in A which makes A into a normed linear space
and which satisfies the multiplicative inequality
Ilxyll S; IIxlillYl1 (x and YEA), (3)
then A is a normed complex algebra. If, in additioQ, A is a complete metric
space relative to this norm, i.e., if A is a Banach space, then we call A a
Banach algebra.
The inequality (3) makes multiplication a continuous operation. This
means that if Xn-+ X and Yn-+ y, then XnYn-+ xy, which follows from (3) and
the identity
xn Yn - xy = (xn - x)Yn + x(yn - y). (4)
Note that we have not required that A be commutative, i.e., that xy = yx
for all x and YEA, and we shall not do so except when explicitly stated.