# length of curve in a metric space

Suppose that $(X,d)$ is a metric space. Let $f$ be a curve, so that
$f:[0,1]\to X$ is a continuous function^{}, and let $$ and
${x}_{i}=f({t}_{i})$ for $0\le i\le n$.
The set $\{{x}_{0},{x}_{1},\mathrm{\dots},{x}_{n}\}$
is called a partition^{} of the curve.
The * of the curve* is defined to be
the supremum over all partitions of the quantity ${\sum}_{i=1}^{n}d({x}_{i},{x}_{i-1})$.

Title | length of curve^{} in a metric space |
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Canonical name | LengthOfCurveInAMetricSpace |

Date of creation | 2013-03-22 16:50:27 |

Last modified on | 2013-03-22 16:50:27 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 8 |

Author | Mathprof (13753) |

Entry type | Definition |

Classification | msc 26B15 |

Defines | length of a curve |