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 x1x2x3, ... of elements of A such that

  1. The sequence zxn converges to the zero element, but
  2. 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

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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