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Exercise 8. Show that a function F: R" → R" preserves the norm || · || if and only if it preserves the inner product (, -).

Exercise 8. Show that a function F: R" → R" preserves the norm || · || if and only if it preserves the inner product (, -).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Exercise 8
Exercise 8. Show that a function F: R" → R" preserves the norm || - || if and only if it preserves the inner product (-, -). 3
Transcribed Image Text:Exercise 8. Show that a function F: R" → R" preserves the norm || - || if and only if it preserves the inner product (-, -). 3
Expert Solution
Step 1

We have the function F:ℝn→ℝn that preserves the norm ·

This implies that this function satisfies all the three axioms of norm

Similarly the function F:ℝn→ℝn that preserves the inner product ·,·

This implies that this function satisfies all the three axioms of inner product

Let v=v1,v2,...,vn,u=u1,u2,...,un∈ℝn,α∈F

we know that any inner product induces the norm given by v=v,v . This is the relation between norm and an inner product. 

 

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