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LSpice
LSpice
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This is a standard result that every rectifiable curve in a metric space admits an arc length parametrization. The proof can be found in many sources. For examples Theorem 3.2 in theseHajłasz - Sobolev spaces on metric-measure spaces notes. Then the arc-length parametrization is defined on $[0,L]$ and a linear change of variables leads us to $g$ defined on $[0,1]$.

This is a standard result that every rectifiable curve in a metric space admits an arc length parametrization. The proof can be found in many sources. For examples Theorem 3.2 in these notes. Then the arc-length parametrization is defined on $[0,L]$ and a linear change of variables leads us to $g$ defined on $[0,1]$.

This is a standard result that every rectifiable curve in a metric space admits an arc length parametrization. The proof can be found in many sources. For examples Theorem 3.2 in Hajłasz - Sobolev spaces on metric-measure spaces. Then the arc-length parametrization is defined on $[0,L]$ and a linear change of variables leads us to $g$ defined on $[0,1]$.

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Piotr Hajlasz
Piotr Hajlasz
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This is a standard result that every rectifiable curve in a metric space admits an arc length parametrization. The proof can be found in many sources. For examples Theorem 3.2 in these notes. Then the arc-length parametrization is defined on $[0,L]$ and a linear change of variables leads us to $g$ defined on $[0,1]$.