Show that the euclidean metric don is a metric as follows: If and , define
(a) Show that x • (y + z) = (x • y) + (x • z).
(b) Show that |x•y| ≤ ||x|| ||y||. [Hint: If , let a = 1/||x|| and b = 1/||y||, and use the fact that ||ax ±by|| ≥ 0.1
(c) Show that ||x + y|| ≤ ||x|| + ||y||. [Hint: Compute (x + y) (x + y) and apply (b).]
(d) Verify that d is a metric.
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