- Show that =
- Prove from the axioms of the inner product the parallelogram identity,
2 = 2
2 + 2
- For a,b ∈ ℝn, define a ⋅ b ≡∑
k=1nβkakbk where βk > 0 for each k. Show this satisfies
the axioms of the inner product. What does the Cauchy Schwarz inequality say in this
- In Problem 3 above, suppose you only know βk ≥ 0. Does the Cauchy Schwarz
inequality still hold? If so, prove it.
- Let f,g be continuous functions and define f ⋅ g ≡∫
dt. Show this satisfies
the axioms of a inner product if you think of continuous functions in the place of a
vector in Fn. What does the Cauchy Schwarz inequality say in this case?
- Show that if f is a real valued continuous function,