b) show that the norms 11·11 and 111:111 artequivalent if and only if sup{||X|||:|1x|| = I}CAD and inf{111X/1) : 11x1) =1^²} 20₁I prove that the notion of bring (quivalentequivalence verlation on thenorms is an367of norms on X 3. Twonormg 11·11 and Ill·lll onаvectorSpace X are equivalent if there existsспо070 such that_ |||X| || = ||X|| = (111 all!, xf X드X

Answered: 3. Two vector norms 11·11 and III'lll…

3. Two vector norms 11·11 and III'lll on a if there exists Space X are equivalent C70 such that |||X| |/ ≤ ||X|| ≤ (|||X|!! xf x.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

b) show that the norms 11·11 and 111:111 art
equivalent if and only if sup{||X|||:|1x|| = I}
CAD and inf{111X/1) : 11x1) =1^²} 20₁
I prove that the notion of bring (quivalent
equivalence verlation on the
norms is an
367
of norms on X

3. Two
normg 11·11 and Ill·lll on
а
vector
Space X are equivalent if there exists
спо
070 such that
_ |||X| || = ||X|| = (111 all!, xf X
드
X

Expert Solution

Step 1

Definition: (Equivalence of Norms)

Two norms on a vector space "V" are said to be equivalent if they define same open subsets of "V" or in other words if they define the same topology on "V".

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