Exercise 1070Topology › Metric and normed spacesL1Added 3 Nov 2026

Product metrics

#product spaces#metric#sup

Statement

Define a metric on $X\times Y$ in terms of metrics on $X$ and $Y$ .

Hint

$d((x_1,y_1),(x_2,y_2))=\max(d_X(x_1,x_2),d_Y(y_1,y_2))$ .

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Definition. $d((x_1,y_1),(x_2,y_2))=\max(d_X(x_1,x_2),d_Y(y_1,y_2))$ .

Properties. Positivity, symmetry, triangle: all inherited component-wise.

Conclusion. $X\times Y$ is a metric space; convergence is component-wise.