vector norm VectorNorm 2001-10-06 03:07:24 2009-04-03 06:10:43 Definition normed vector space Euclidean norm \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} A vector norm on the real vector space $V$ is a function $f : V \to \mathbb{R}$ that satisfies the following properties: \begin{eqnarray*} f(x) = 0 \iff x = 0 && \\ f(x) \ge 0 && x \in V \\ f(x+y) \leq f(x)+f(y) && x,y \in V \\ f(\alpha x) = |\alpha|f(x) && \alpha \in \mathbb{R},x\in V \end{eqnarray*} 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 \emph{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 {\em normed vector space}. Given a metric on the vector space, a necessary and sufficient condition for this space to be a normed space, is \begin{eqnarray*} d(x+a,y+a)=&d(x,y) & \forall x,y,a \in V\\ d(\alpha x,\alpha y)=&|\alpha|d(x,y) &\forall x,y \in V, \alpha \in \mathbb{R}\\ \end{eqnarray*} 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.