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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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}
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