(b) Show that x. y < ||x|| ||y||- [Hint: If x, y # 0, let a = , 6= and use the fact that ||ax + by | 2 0.] (This is known as the Cauchy-Schwarz Inequality) (c) Show that ||x+y|| < ||x||+||y||. [Hint: Compute (x+y) · (x+y) and apply part (b).] (d) Show that d is a metric.

(b) Show that x. y < ||x|| ||y||- [Hint: If x, y # 0, let a = , 6= and use the fact that ||ax + by | 2 0.] (This is known as the Cauchy-Schwarz Inequality) (c) Show that ||x+y|| < ||x||+||y||. [Hint: Compute (x+y) · (x+y) and apply part (b).] (d) Show that d is a metric.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.4: Ordered Integral Domains

Problem 8E: If x and y are elements of an ordered integral domain D, prove the following inequalities. a....

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Question

(b) Show that x. y < ||x|| ||y||- [Hint: If x, y # 0, let a = , 6= and use the fact
that ||ax + by | 2 0.] (This is known as the Cauchy-Schwarz Inequality)
(c) Show that ||x+y|| < ||x||+||y||. [Hint: Compute (x+y) · (x+y) and apply part (b).]
(d) Show that d is a metric.

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