# 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 $0=t_{0} and $x_{i}=f(t_{i})$ for $0\leq i\leq n$. The set $\{x_{0},x_{1},\ldots,x_{n}\}$ 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 LengthOfCurveInAMetricSpace 2013-03-22 16:50:27 2013-03-22 16:50:27 Mathprof (13753) Mathprof (13753) 8 Mathprof (13753) Definition msc 26B15 length of a curve