Normed vector space

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In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial.

1. The zero vector, 0, has zero length; every other vector has a positive length.

2. Multiplying a vector by a positive number changes its length without changing its direction. Moreover,

3. The triangle inequality holds. That is, taking norms as distances, the distance from point A through B to C is never shorter than going directly from A to C, or the shortest distance between any two points is a straight line.

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 vector space[1]. Normed vector spaces are central to the study of linear algebra and functional analysis.

Contents

Definition

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

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

Topological structure

If (V, ‖·‖) is a normed vector space, the norm ‖·‖ induces a notion of distance and therefore a topology on V. This distance is defined in the natural way: the distance between two vectors u and v is given by ‖uv‖. This topology is precisely the weakest topology which makes ‖·‖ continuous and which is compatible with the linear structure of V in the following sense:

Similarly, for any semi-normed vector space we can define the distance between two vectors u and v as ‖uv‖. This turns the seminormed space into a Pseudo-metric space (notice this is weaker than a metric) and allows the definition of notions such as continuity and convergence. To put it more abstractly every semi-normed vector space is a topological vector space and thus carries a topological structure which is induced by the semi-norm.

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