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 space
of any dimension
Vectors 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 by
An 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 of
matrices
with norm
is a normed linear space.
The vector space of all infinite sequences
of real numbers satisfying the convergence condition
with norm defined by
is 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.