M1 := sup ||T(x)|| and ||x||=1 M2 := inf{C > 0 : ||T(x)|| < C||x|| for all x e R"}. a) Prove that M1 < ||T||. b) Using the linear property of T, prove that if x + 0, then ||T(x)|| < M1. |||| c) Prove that M1 = M2 = ||T||. %3D

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 56E
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8.2.11. Fix T e L(R"; R"). Set
M1 := sup ||T(x)|| and
||x||=1
M2 := inf{C > 0 : ||T(x)|| < C||x|| for all x e R"}.
a) Prove that M1 < ||T||.
b) Using the linear property of T, prove that if x + 0, then
||T(x)||
< M1.
||||
c) Prove that M1 = M2 = ||T||.
Transcribed Image Text:8.2.11. Fix T e L(R"; R"). Set M1 := sup ||T(x)|| and ||x||=1 M2 := inf{C > 0 : ||T(x)|| < C||x|| for all x e R"}. a) Prove that M1 < ||T||. b) Using the linear property of T, prove that if x + 0, then ||T(x)|| < M1. |||| c) Prove that M1 = M2 = ||T||.
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