If (X, d_{1}) and (Y, d_{2}) are metric spaces, we define d: (X × Y ) × (X × Y ) \rightarrow R by d((x_{1}, y_{ 1}),(x_{2}, y_{2})) = d_{1}(x, x′) + d_{2}(y, y′). Answer the following literals a) Prove that d is a metric b) If X = Y = R with d1 equal to the usual metric and d2 the discrete metric. Describe what the balls are like with the metric d on R^{2}

If (X, d_{1}) and (Y, d_{2}) are metric spaces, we define d: (X × Y ) × (X × Y ) \rightarrow R by d((x_{1}, y_{ 1}),(x_{2}, y_{2})) = d_{1}(x, x′) + d_{2}(y, y′). Answer the following literals a) Prove that d is a metric b) If X = Y = R with d1 equal to the usual metric and d2 the discrete metric. Describe what the balls are like with the metric d on R^{2}

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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