vector norm

A vector norm  on the real vector space $V$ is a function $f:V\to\mathbb{R}$ that satisfies the following properties:

 $\displaystyle f(x)=0\iff x=0$ $\displaystyle f(x)\geq 0$ $\displaystyle x\in V$ $\displaystyle f(x+y)\leq f(x)+f(y)$ $\displaystyle x,y\in V$ $\displaystyle f(\alpha x)=|\alpha|f(x)$ $\displaystyle\alpha\in\mathbb{R},x\in V$

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 $||\,\cdot\,||$ is a vector $x$ satisfying $||\,x\,||=1$.

A vector norm on a complex vector space is defined similarly.

A common (and useful) example of a real norm is the Euclidean norm given by $||x||=(x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2})^{1/2}$ defined on $V=\mathbb{R}^{n}$. 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

 $\displaystyle d(x+a,y+a)=$ $\displaystyle d(x,y)$ $\displaystyle\forall x,y,a\in V$ $\displaystyle d(\alpha x,\alpha y)=$ $\displaystyle|\alpha|d(x,y)$ $\displaystyle\forall x,y\in V,\alpha\in\mathbb{R}$

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