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1. Consider the space of absolutely convergent sequences equipped with the following norm 1/2 00 I| (¤1, .., Tn, ..) |1,2= #;|+ (0.1) j=1 j=1 a). Prove that || - ||1.2 is a norm. b). Is this norm equivalent to the standard || r ||1= ET;[? 1/2 c). Is it equivalent to || | ||2= ( E;=1 T;|²

1. Consider the space of absolutely convergent sequences equipped with the following norm 1/2 00 I| (¤1, .., Tn, ..) |1,2= #;|+ (0.1) j=1 j=1 a). Prove that || - ||1.2 is a norm. b). Is this norm equivalent to the standard || r ||1= ET;[? 1/2 c). Is it equivalent to || | ||2= ( E;=1 T;|²

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.1: Infinite Sequences And Summation Notation
Problem 72E
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1. Consider the space of absolutely convergent sequences equipped with the following norm 1/2 00 (0.1) j=1 j=1 a). Prove that || - ||1.2 is a norm. b). Is this norm equivalent to the standard || ||1=E *;|? 1/2 c). Is it equivalent to || | ||2= (E ? ) ?
Transcribed Image Text:1. Consider the space of absolutely convergent sequences equipped with the following norm 1/2 00 (0.1) j=1 j=1 a). Prove that || - ||1.2 is a norm. b). Is this norm equivalent to the standard || ||1=E *;|? 1/2 c). Is it equivalent to || | ||2= (E ? ) ?
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