Past exam of the metric spaces and complex analysis course of the University of Oxford / 2014 / 2 / a / i / Solution

Nonnegativity and symmetry follow from the corresponding metric axioms in the two factors. The maximum is zero exactly when both coordinate distances are zero, hence exactly when the ordered pairs are equal.

For the triangle inequality, put

p = (m_{1}, n_{1})

,

q = (m_{2}, n_{2})

and

r = (m_{3}, n_{3})

. Then

δ (p, r) \leq max {d_{M} (m_{1}, m_{2}) + d_{M} (m_{2}, m_{3}), d_{N} (n_{1}, n_{2}) + d_{N} (n_{2}, n_{3})} \leq max {d_{M} (m_{1}, m_{2}), d_{N} (n_{1}, n_{2})} + max {d_{M} (m_{2}, m_{3}), d_{N} (n_{2}, n_{3})} = δ (p, q) + δ (q, r) .

Therefore the maximum formula defines a metric, the maximum product metric.

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