normed vector space
Let be a field which is either or . A over is a pair where is a vector space over and is a function such that
- 1.
for all and if and only if in (positive definiteness)
- 2.
for all and all
- 3.
for all (the triangle inequality
)
The function is called a norm on .
Some properties of norms:
- 1.
If is a subspace
of then can be made into a normed space
by simply restricting the norm on to . This is called the induced norm on .
- 2.
Any normed vector space is a metric space under the metric given by . This is called the metric induced by the norm .
- 3.
It follows that any normed space is a locally convex topological vector space, in the topology
induced by the metric defined above.
- 4.
In this metric, the norm defines a continuous map
from to - this is an easy consequence of the triangle inequality.
- 5.
If is an inner product space
, then there is a natural induced norm given by for all .
- 6.
The norm is a convex function of its argument.
Title | normed vector space |
Canonical name | NormedVectorSpace |
Date of creation | 2013-03-22 12:13:45 |
Last modified on | 2013-03-22 12:13:45 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 14 |
Author | rspuzio (6075) |
Entry type | Definition |
Classification | msc 46B99 |
Synonym | normed space |
Synonym | normed linear space |
Related topic | CauchySchwarzInequality |
Related topic | VectorNorm |
Related topic | PseudometricSpace |
Related topic | MetricSpace |
Related topic | UnitVector |
Related topic | ProofOfGramSchmidtOrthogonalizationProcedure |
Related topic | EveryNormedSpaceWithSchauderBasisIsSeparable |
Related topic | EveryNormedSpaceWithSchauderBasisIsSeparable2 |
Related topic | FrobeniusProduct |
Defines | norm |
Defines | metric induced by a norm |
Defines | metric induced by the norm |
Defines | induced norm |