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norm

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Definition

For kk a field equipped with a valuation (most usually, a local field such as \mathbb{R}, \mathbb{C}, or a p-adic completion of a number field), a norm on a kk-vector space VV is a function

:V {\vert-\vert} : V \to \mathbb{R}

such that for all λk\lambda \in k, v,wVv,w \in V we have

  1. λv=λv{\vert \lambda v \vert} = {\vert \lambda\vert} {\vert v \vert} (where λ\vert \lambda \vert denotes the valuation)

  2. v+wv+w{\vert v + w\vert } \leq {\vert v \vert } + {\vert w \vert} (“triangle inequality?”)

  3. if v=0{\vert v\vert} = 0 then v=0v = 0.

If the third property is not required, one speaks of a seminorm.

If the triangle identity is strengthened to

  • v+wmax(v,w){\vert v + w\vert } \leq max ({\vert v\vert}, {\vert w\vert})

one speaks of a non-archimedean seminorm.

A vector space equipped with a norm is a normed vector space.

Each seminorm determines a topology, which is Hausdorff precisely if it is a norm.

A topological vector space is called (semi-)normed if its topology can be induced by a (semi-)norm.

Two seminorms 1{\vert - \vert}_1 and 2{\vert - \vert}_2 are called equivalent if there are 0<C,C0 \lt C, C' \in \mathbb{R} such that for all vv we have

Cv 1v 2Cv 1. C {\vert v \vert}_1 \leq {\vert v \vert}_2 \leq C' {\vert v \vert}_1 \,.

Equivalent seminorms determine the same topology.

The collection of (bounded) multiplicative seminorms on a (Banach) ring is called its analytic spectrum (see there for details).

Examples

  • The standard absolute value is a norm on the real numbers.

  • More generally, on any Cartesian space n\mathbb{R}^n the Euclidean norm is given by

    (x 1,,x n)( i=1 n(x i) 2). (x^1, \cdots, x^n) \mapsto \sqrt(\sum_{i=1}^n (x^i)^2) \,.

Minkowski Functionals

Let VV be a vector space and BVB \subseteq V an absorbing absolutely convex subset. The Minkowski functional of BB is the function μ B:V\mu_B \colon V \to \mathbb{R} defined by:

μ B(v)=inf{t>0:vtB} \mu_B(v) = \inf\{t \gt 0 : v \in t B\}

This is a semi-norm on VV.

Revised on May 23, 2013 18:41:23 by Andrew Stacey (192.76.7.217)