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show that d is a metric on a map D(x,y):=2d(x,y)

Linear Algebra
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i dont know about metric space but JamesJ does i hope this help
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what is D(x,y) ??
you can't define a metric by just just a function without any rule. you might have different rules that might not be metric.

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Other answers:

you use the definition of a metric spaces together with their conditions,and they should all hold for it to be a metric space.

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