The solution of this problem in the question bank does not seem to have used the fact that ||x|| = <x,-x>1/2 Let X be an inner product space with the inner product given by <, >. For x € X,define the function ||-|| : X → K given by ||x|| = < x, − x > 1/2, the non negativesquare root of < x, x>. Show that ||· || :X → K defines a norm on X and|< (x, y) >I ≤|x || ||y|| for all x, ye X. Also show that for all x, ye X,|x + y² + x - y = 2x1² +|y||²)

Given : x=x , -x12 or x2 = x, -x To Prove : (a) ·:X→K is a norm. : (b)…

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The solution of this problem in the question bank does not seem to have used the fact that ||x|| = 1/2

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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The solution of this problem in the question bank does not seem to have used the fact that ||x|| = <x,-x>^1/2

Let X be an inner product space with the inner product given by <, >. For x € X,
define the function ||-|| : X → K given by ||x|| = < x, − x > 1/2, the non negative
square root of < x, x>. Show that ||· || :X → K defines a norm on X and
|< (x, y) >I ≤|x || ||y|| for all x, ye X. Also show that for all x, ye X,
|x + y² + x - y = 2x1² +|y||²)

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