Normed Vector Space

A normed vector space is a pair (V, ‖·‖ ) where V is a vector space and ‖·‖ a norm on V.

A seminormed vector space is a pair (V,p) where V is a vector space and p a seminorm on V.

We often omit p or ‖·‖ and just write V for a space if it is clear from the context what (semi) norm we are using.

In a more general sense, a vector norm can be taken to be any real-valued function that satisfies these three properties. The properties 1. and 2. together imply that

if and only if .

A useful variation of the triangle inequality is

for any vectors x and y.

This also shows that a vector norm is a continuous function.

Read more about Normed Vector Space:  Topological Structure, Linear Maps and Dual Spaces, Normed Spaces As Quotient Spaces of Seminormed Spaces, Finite Product Spaces

Famous quotes containing the word space:

    Finally she grew quiet, and after that, coherent thought. With this, stalked through her a cold, bloody rage. Hours of this, a period of introspection, a space of retrospection, then a mixture of both. Out of this an awful calm.
    Zora Neale Hurston (1891–1960)