36 CHAPTER 1. PRELIMINARIES

3. For a,b ∈ Rn, define a · b ≡∑n

k=1 βkakbk where βk > 0 for each k. Show this satisfiesthe axioms of the inner product. What does the Cauchy Schwarz inequality say inthis case.

4. In Problem 3 above, suppose you only know βk ≥ 0. Does the Cauchy Schwarz in-equality still hold? If so, prove it.

5. Let f, g be continuous functions and define f · g ≡∫ 1

0f (t) g (t)dt. Show this satisfies

the axioms of a inner product if you think of continuous functions in the place of avector in Fn. What does the Cauchy Schwarz inequality say in this case?

6. Show that if f is a real valued continuous function,(∫ b

af (t) dt

)2≤ (b− a)

∫ b

af (t)

2dt.