## Normed Linear Spaces

In one, two or three dimensions, the length of a vector is intuitively given by the Euclidean metric and can easily be extended to any Euclidean spaceof any dimensionVectors in have the following properties:

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

2. Multiplying a vector by a scalar changes its length by the magnitude of that scalar:

for any scalar
3. The triangle inequality holds. That is, taking norms as distances, the distance from point A to C via B is at least as long as the distance from A to C direct:

Alternatively, the shortest distance between any two points is a straight line.

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

Examples:

Every inner product of an inner product space determines the norm given byAn inner product space is always a normed linear space with the inner product norm. However, a normed linear space is not necessarily an inner product space.

The vector space ofmatriceswith normis a normed linear space.

The vector space of all infinite sequencesof real numbers satisfying the convergence conditionwith norm defined byis a normed linear space.

To prove 3. We need Minkowski's inequality

Every norm induces a metric, so a normed linear space is also a metric space.