A Fundamental Theorem for Linear Operators. The Neumann Series.
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Starting with this story about Banach’s fix-point theorem, we saw its endless applications across various disciplines in mathematics. In this story, I want to follow-up with this by looking at yet another fundamental construct that is strongly related to this theorem.
Although, the story is written as abstract as possible in order to provide a generic as possible point of view, it should suffice to be familiar with the material given in one of my previous stories about norms (see here).
The example above shows the usefulness about the convergence criterion for the Neumann series. Although we could have applied the Banach fix-point theorem directly onto the Volterra integral operator
A, we would have had to make further restrictive assumptions on
K in order to make
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