Normed linear space

In mathematics, a normed vector space is a vector space on which a norm is defined. In a vector space with 1- 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 ${\displaystyle \mathbb {R} ^{n}}$. 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 ${\displaystyle (V,\|\cdot \|)}$ where ${\displaystyle V}$ is a vector space and ${\displaystyle \|\cdot \|}$ a norm on ${\displaystyle V}$.

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