State whether the following statements are true or false. Justify with a short proof or counter example. The image of a Cauchy sequence under a bounded linear map is also a Couchy sequence. If A is a bounded linear operator on a Hilbert space such that AA* = I, then A'A=I

State whether the following statements are true or false. Justify with a short proof or counter example. The image of a Cauchy sequence under a bounded linear map is also a Couchy sequence. If A is a bounded linear operator on a Hilbert space such that AA* = I, then A'A=I

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.4: Prime Factors And Greatest Common Divisor

Problem 17E

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Question

1.
State whether the following statements are true or false. Justify with a short proof or a
counter example.
The image of a Cauchy sequence under a bounded linear map is also a Couchy
sequence.
If A is a bounded linear operator on a Hilbert space such that AA* = I, then
A*A=I

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Follow-up Question

The second statement of true and false is not clear. Request explain again. Statement does not say that A is an adjoint operator, so how do we use it in second equality? if we start with <A*x,A*x> do we get

<A*x,A*x> = <AA*x,x> = <x,x> = ||x|| squared?

can we proceed from this??

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