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.
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.