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The solution of this problem in the question bank does not seem to have used the fact that ||x|| = 1/2

The solution of this problem in the question bank does not seem to have used the fact that ||x|| = 1/2

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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The solution of this problem in the question bank does not seem to have used the fact that ||x|| = <x,-x>1/2

Let X be an inner product space with the inner product given by <, >. For x € X, define the function ||-|| : X → K given by ||x|| = < x, − x > 1/2, the non negative square root of < x, x>. Show that ||· || :X → K defines a norm on X and |< (x, y) >I ≤|x || ||y|| for all x, ye X. Also show that for all x, ye X, |x + y² + x - y = 2x1² +|y||²)
Transcribed Image Text:Let X be an inner product space with the inner product given by <, >. For x € X, define the function ||-|| : X → K given by ||x|| = < x, − x > 1/2, the non negative square root of < x, x>. Show that ||· || :X → K defines a norm on X and |< (x, y) >I ≤|x || ||y|| for all x, ye X. Also show that for all x, ye X, |x + y² + x - y = 2x1² +|y||²)
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