Let N be a linear space and d be a metric on N such that d(x...

 

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Let N be a linear space and d be a metric on N such that d(x + z, y + z) = d(x, y), x,y €N and d(ax, ay) = |ald(x,y), a E F. Show that ||. ||: N - R defined by: I|x|| = d(x, 0), x EN, is a norm on N. Let N be a linear space and d be a metric on N such that d(x + z, y + z) = d(x, y), x,y €N and d(ax, ay) = |ald(x,y), a E F. Show that ||. ||: N - R defined by: I|x|| = d(x, 0), x EN, is a norm on N.