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| Title of object: |
vector norm |
| Canonical Name: |
VectorNorm |
| Type: |
Definition |
| Created on: |
2001-10-06 03:07:24 |
| Modified on: |
2003-08-01 08:25:29 |
| Classification: |
msc:46B20 |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts} |
Content:
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 common (and useful) example is the Euclidean norm given by $||x||=(x_1^2 + x_2^2 + \cdots + x_n^2)^{1/2}$ defined on $V=\mathbb{R}^n$.
Also note that there does not exist any norm on all metric spaces; when it does, the space is called a \emph{normed vector space}. |
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