Normed space

In mathematics, a normed vector space is a vector space on which a norm is defined. In a vector space with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive. This intuition can easily be extended to any real vector space Rⁿ. The length of a vector in such a vector space has the following properties:

The generalization of these three properties to more abstract vector spaces leads to the notion of norm. A vector space on which a norm is defined is then called a normed space or normed vector space. Normed vector spaces are central to the study of linear algebra and functional analysis.

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

A useful variation of the triangle inequality is

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

Note that property 2 depends on a choice of norm ${\displaystyle |\alpha |}$ on the field of scalars. When the scalar field is ${\displaystyle \mathbb {R} }$ (or more generally a subset of ${\displaystyle \mathbb {C} }$), this is usually taken to be the ordinary absolute value, but other choices are possible. For example, for a vector space over ${\displaystyle \mathbb {Q} }$ one could take ${\displaystyle |\alpha |}$ to be the p-adic norm, which gives rise to a different class of normed vector spaces.

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