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length of curve in a metric space (Definition)

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 < t_1 < \cdots < t_n=1$ and $ x_i = f(t_i)$ for $ 0 \le i \le n$. The set $ \{x_0, x_1, \ldots , x_n\}$ is called a partition of the curve. The length 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})$.



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Also defines:  length of a curve
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Cross-references: supremum, partition, continuous function, curve, metric space
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This is version 5 of length of curve in a metric space, born on 2007-03-16, modified 2007-07-21.
Object id is 9085, canonical name is LengthOfCurveInAMetricSpace.
Accessed 931 times total.

Classification:
AMS MSC: 26B15 (Real functions :: Functions of several variables :: Integration: length, area, volume)
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