In mathematics, an element z of a Banach algebra A is called a topological divisor of zero if there exists a sequence x1, x2, x3, ... of elements of A such that
The sequence zxn converges to the zero element, but
The sequence xn does not converge to the zero element.
If such a sequence exists, then one may assume that ||xn|| = 1 for all n.
If A is not commutative, then z is called a left topological divisor of zero, and one may define right topological divisors of zero similarly.
Examples
If A has a unit element, then the invertible elements of A form an open subset of A, while the non-invertible elements are the complementary closed subset. Any point on the boundary between these two sets is both a left and right topological divisor of zero.
In particular, any quasinilpotent element is a topological divisor of zero (e.g. the Volterra operator).
An operator on a Banach space X, which is injective, not surjective, but whose image is dense in X, is a left topological divisor of zero.
Generalization
The notion of a topological divisor of zero may be generalized to any topological algebra. If the algebra in question is not first-countable, one must substitute nets for the sequences used in the definition.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
