Non-Archimedean Operator Theory by Toka Diagana & François Ramaroson

Author:Toka Diagana & François Ramaroson
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Let be a linear mapping. Clearly, and hence there exists for i, j = 1, 2, …, n such that

In what follows, we show that the arbitrary linear operator A given above is necessarily bounded. Indeed, for all

where

Consequently, is a bounded linear operator.

Example 3.3.

Let where p ≥ 2 is a prime and let be the space of continuous functions from into , which we equip with its sup-norm given by for all . Let , be a polynomial of degree N with coefficients a 1, a 2, …, a N belonging to . Consider the so-called multiplication operator defined by, A(f)(z) = P(z)f(z) for all and . Clearly, A is a linear operator. Further, for all where . Therefore, A is a bounded linear operator.

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