vector norm (original) (raw)
| f(x)=0⇔x=0 | |
|---|---|
| f(x)≥0 | x∈V |
| f(x+y)≤f(x)+f(y) | x,y∈V |
| f(αx)=|α | f(x) |
Such a function is denoted as ||x||. Particular norms are distinguished by subscripts, such as ||x||V, when referring to a norm in the space V. A unit vector
with respect to the norm ||⋅|| is a vector x satisfying||x||=1.
A common (and useful) example of a real norm is the Euclidean norm given by ||x||=(x12+x22+⋯+xn2)1/2 defined on V=ℝn. Note, however, that there exists vector spaces
which are metric, but upon which it is not possible to define a norm. If it possible, the space is called a normed vector space. Given a metric on the vector space, a necessary and sufficient condition for this space to be a normed space, is
| d(x+a,y+a)= | d(x,y) | ∀x,y,a∈V |
|---|---|---|
| d(αx,αy)= | |α | d(x,y) |
But given a norm, a metric can always be defined by the equation d(x,y)=||x-y||. Hence every normed space is a metric space.
| Title | vector norm |
|---|---|
| Canonical name | VectorNorm |
| Date of creation | 2013-03-22 11:43:00 |
| Last modified on | 2013-03-22 11:43:00 |
| Owner | mike (2826) |
| Last modified by | mike (2826) |
| Numerical id | 30 |
| Author | mike (2826) |
| Entry type | Definition |
| Classification | msc 46B20 |
| Classification | msc 18-01 |
| Classification | msc 20H15 |
| Classification | msc 20B30 |
| Related topic | Vector |
| Related topic | Metric |
| Related topic | Norm |
| Related topic | VectorPnorm |
| Related topic | NormedVectorSpace |
| Related topic | MatrixNorm |
| Related topic | MatrixPnorm |
| Related topic | FrobeniusMatrixNorm |
| Related topic | CauchySchwarzInequality |
| Related topic | MetricSpace |
| Related topic | VectorSpace |
| Related topic | LpSpace |
| Related topic | OperatorNorm |
| Related topic | BoundedOperator |
| Related topic | SemiNorm |
| Related topic | BanachSpace |
| Related topic | HilbertSpace |
| Related topic | UnitVector |
| Defines | normed vector space |
| Defines | Euclidean norm |