3.Two norms |||| and ||| ||| on avector space are equivalent if thereexists c>o such that// | ||||| x |||≤|x|| ≤ c |||*|||,xe t(a) Show that the norms || || and |||.I are equivalent if and only ifthe identity map of is bicon-tinuousbetweenthe.||-||-topology of|||-topology of *.and the |||(b) Show that the norms |||| and||| ||| are equivalent if and only.if sup{|||* ||| || x || = 1} <∞ andinf{||| x ||| ||x|| = 1} > 0.(c) Prove that the notion of beingequivalent norms is an equiv-alence relation on the set ofnorms on *.

Given: In this question, the given details are, There are two norms are present, · and · on the…

- C 3. Two norms |||| and ||| ||| on a vector space are equivalent if there exists c o such that ||| x ||| ≤|x|| ≤ c ||x, xe t Show that the norms || || and ||| • ||| are equivalent if and only if the identity map of is bicon- tinuous between the (a) ||-||-topology of |||-topology of *. (b) Show that the norms || || and ||| ||| are equivalent if and only if sup{|||* ||| || x || = 1} <∞ and inf{|||*|||||||=1} > 0. (c) Prove that the notion of being equivalent norms is an equiv- alence relation on the set of and the ||| norms on .

- C 3. Two norms |||| and ||| ||| on a vector space are equivalent if there exists c o such that ||| x ||| ≤|x|| ≤ c ||x, xe t Show that the norms || || and ||| • ||| are equivalent if and only if the identity map of is bicon- tinuous between the (a) ||-||-topology of |||-topology of . (b) Show that the norms || || and ||| ||| are equivalent if and only if sup{||| ||| || x || = 1} <∞ and inf{|||*|||||||=1} > 0. (c) Prove that the notion of being equivalent norms is an equiv- alence relation on the set of and the ||| norms on .

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

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3.
Two norms |||| and ||| ||| on a
vector space are equivalent if there
exists c>o such that
// | ||
||| x |||≤|x|| ≤ c |||*|||,
xe t
(a) Show that the norms || || and |||
.
I are equivalent if and only if
the identity map of is bicon-
tinuous
between
the
.
||-||-topology of
|||-topology of *.
and the |||
(b) Show that the norms |||| and
||| ||| are equivalent if and only
.
if sup{|||* ||| || x || = 1} <∞ and
inf{||| x ||| ||x|| = 1} > 0.
(c) Prove that the notion of being
equivalent norms is an equiv-
alence relation on the set of
norms on *.

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