Let (an)nez be a sequence of real numbers. Recall the 2(N), as a vector space over...

Transcribed Image Text:

Let (an)nez be a sequence of real numbers. Recall the 2(N), as a vector space over R, has the norm given by 2||(alle) (n=1 lan2)1/2 = lim N-(N=121/2 We will often denote this by llanlle to have a cleaner notation. (a) Let (a) and (b) E2(N). Prove the Cauchy-Schwarz inequality. K(an), (b) llanlle balle, which in this inner product space, takes the form: Enabol (Elanl²)1/2 (2)1/2 (b) Prove that || 2 satisfies the triangle inequality. (Once we have this fact, it's now easy to see that ² is a normed linear space.) Let (an)nez be a sequence of real numbers. Recall the 2(N), as a vector space over R, has the norm given by 2||(alle) (n=1 lan2)1/2 = lim N-(N=121/2 We will often denote this by llanlle to have a cleaner notation. (a) Let (a) and (b) E2(N). Prove the Cauchy-Schwarz inequality. K(an), (b) llanlle balle, which in this inner product space, takes the form: Enabol (Elanl²)1/2 (2)1/2 (b) Prove that || 2 satisfies the triangle inequality. (Once we have this fact, it's now easy to see that ² is a normed linear space.)

Answer rating: 100% (QA)

a Proof of CauchySchwarz Inequality Lets denote the inner product of vectors an and bn as langle a b ... View the full answer