Topological divisor of zero
- 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.
- 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 , which is injective, not surjective, but whose image is dense in , is a left topological divisor of zero.
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.
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