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Ex3: Let X be a normed linear space. Use the Hahn-Banach theorem to prove the following statements. i) For any non zero x in X, there is a bounded linear functional in X such that ||||= 1 and (x) = ||x||. ii) If x, y in X verify 4(x) = (y) in X', then x = y.

Ex3: Let X be a normed linear space. Use the Hahn-Banach theorem to prove the following statements. i) For any non zero x in X, there is a bounded linear functional in X such that ||||= 1 and (x) = ||x||. ii) If x, y in X verify 4(x) = (y) in X', then x = y.

Advanced Engineering Mathematics
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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Ex3: Let X be a normed linear space. Use the Hahn-Banach theorem to prove the following statements. For any non zero x in X, there is a bounded linear functional O in X' such that ll|| = 1 and o(x)= ||x||- i) %3D ii) If x, y in X verify p(x) = 4p(y) ▼ p in x' , then x = y.
Transcribed Image Text:Ex3: Let X be a normed linear space. Use the Hahn-Banach theorem to prove the following statements. For any non zero x in X, there is a bounded linear functional O in X' such that ll|| = 1 and o(x)= ||x||- i) %3D ii) If x, y in X verify p(x) = 4p(y) ▼ p in x' , then x = y.
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