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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. ∗ hNormedAlgebrai created: h2013-03-21i by: hCWooi version: h38286i Privacy setting: h1i hDefinitioni h46H05i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 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. 2