# Topological divisor of zero

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In mathematics, an element *z* of a Banach algebra *A* is called a **topological divisor of zero** if there exists a sequence *x*_{1}, *x*_{2}, *x*_{3}, ... of elements of *A* such that

- The sequence
*zx*_{n}converges to the zero element, but - The sequence
*x*_{n}does not converge to the zero element.

If such a sequence exists, then one may assume that ||*x*_{n}|| = 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 , which is injective, not surjective, but whose image is dense in , 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.

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