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).

Application:

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 A non-expansive.

Thanks for reading!