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Normes (results part) #ALNT-LB 1.1.R Chapter 1 Section 1 #Algebraic Number Theory # Lecture note Benois
Definition A norm over a field K.
In particular, non-Archimedean norm.
Proposition 1.1 (criterion of a function being a norm) ||-|| is a norm if and only if ||1+x|| leq 2 for any ||x|| leq 1.
Proposition 1.2 (criterion of a norm being non-Archimedean) A norm is non-Archimedean if and only if ||n|| leq 1 for any n natural number.
Definition The distance associated to a normed field K.
Proposition 1.3 Certain maps are continuous.
Definition Two norms being equivalent.
Proposition 1.3 (criterion of two norms being equivalent)
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By LucDefinition A norm over a field K.
In particular, non-Archimedean norm.
Proposition 1.1 (criterion of a function being a norm) ||-|| is a norm if and only if ||1+x|| leq 2 for any ||x|| leq 1.
Proposition 1.2 (criterion of a norm being non-Archimedean) A norm is non-Archimedean if and only if ||n|| leq 1 for any n natural number.
Definition The distance associated to a normed field K.
Proposition 1.3 Certain maps are continuous.
Definition Two norms being equivalent.
Proposition 1.3 (criterion of two norms being equivalent)