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normed algebra∗
CWoo†
2013-03-21 21:04:49
A ring A is said to be a normed ring if A possesses a norm k · k, that is, a
non-negative real-valued function k · k : A → R such that for any a, b ∈ A,
1. kak = 0 iff a = 0,
2. ka + bk ≤ kak + kbk,
3. k − ak = kak, and
4. kabk ≤ kakkbk.
Remarks.
• If A contains the multiplicative identity 1, then 0 < k1k ≤ k1kk1k and so
1 ≤ k1k.
• However, it is usually required that in a normed ring, k1k = 1.
• k · k defines a metric d on A given by d(a, b) = ka − bk, so that A with
d is a metric space and one can set up a topology on A by defining its
subbasis a collection of B(a, r) := {x ∈ A | d(a, x) < r} called open balls
for any a ∈ A and r > 0. With this definition, it is easy to see that k · k
is continuous.
• Given a sequence {an } of elements in A, we say that a is a limit point of
{an }, if
lim kan − ak = 0.
n→∞
By the triangle inequality, a, if it exists, is unique, and so we also write
a = lim an .
n→∞
• In addition, the last condition ensures that the ring multiplication is continuous.
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An algebra A over a field k is said to be a normed algebra if
1. A is a normed ring with norm k · k,
2. k is equipped with a valuation | · |, and
3. kαak = |α|kak for any α ∈ k and a ∈ A.
Remarks.
• Alternatively, a normed algebra A can be defined as a normed vector space
with a multiplication defined on A such that multiplication is continuous
with respect to the norm k · k.
• Typically, k is either the reals R or the complex numbers C, and A is called
a real normed algebra or a complex normed algebra correspondingly.
• A normed algebra that is complete with respect to the norm is called
Banach algebra (the underlying field must be complete and algebraically
closed), paralleling with the analogy with a Banach space versus a normed
vector space.
• Normed rings and normed algebras are special cases of the more general
notions of a topological ring and a topological algebra, the latter of which is
defined as a topological ring over a field such that the scalar multiplication
is continuous.
References
[1] M. A. Naimark: Normed Rings, Noordhoff, (1959).
[2] C. E. Rickart: General Theory of Banach Algebras, Van Nostrand, 1960.
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