proof of Neumann series in Banach algebras
Let x be an element of a Banach algebra with identity, . By applying the properties of the Norm in a Banach algebra, we see that the partial sums form a Cauchy sequence: for (as is well known from real analysis), so by completeness of the Banach Algebra, the series converges to some element .
We observe that for any ,
Furthermore, , so .
Thus, by taking the limit on both sides of (1), we get
(We can exchange the limit with the multiplication by , since the multiplication in Banach algebras is continuous)
Since the Banach algebra generated by a single element is commutative and and are both in the Banach algebra generated by , we also get . Hence, .
As in the first paragraph, the last claim again follows by applying the geometric series for real numbers.
|Title||proof of Neumann series in Banach algebras|
|Date of creation||2013-03-22 17:32:40|
|Last modified on||2013-03-22 17:32:40|
|Last modified by||FunctorSalad (18100)|