T E C H N I S C H E UNIVERSIT ¨ AT DARMSTADT

Fachbereich Mathematik Prof. Dr. W. Trebels Dr. V. Gregoriades Dr. A. Linshaw

T E C H N I S C H E UNIVERSIT ¨ AT DARMSTADT

A

3rd Tutorial Analysis II (engl.)

Summer Semester 2010

(T3.1)

Let X be an infinite set. For x and y in X, define d(x, y) = 0 if x=y and d(x, y) = 1 if x6=y.

1. Show that dis a metric on X, and describe the-ballU(x) for any x∈X and >0.

2. Which subsets of X are closed? Which are open?

(T3.2)

We consider the “p-norms” from Example VI.1.5. Prove that it is really the case that the mappingk · kp :Rⁿ →R, 1≤p≤ ∞, defines a norm on Rⁿ.

Hint: Use H¨older’s inequality Pn

i=|x_iy_i| ≤ kxk_pkyk_q, for p, q > 1 satisfying ¹_p + ¹_q = 1.

(Cor. IV.2.16, Analysis I script).

Prove also that the unit ball B_p(0,1) := {x ∈ Rⁿ : kxk_p < 1} is convex for each 1≤ p≤ ∞. Can one in a similar way prove that the unit ball is convex for any norm on a vector space over R?

(A subsetA of a vector space overRis calledconvex if for allx, y ∈Aand allλ∈[0,1]

we have λx+ (1−λ)y∈A.)

(T3.3) (An application of Parseval’s Formula) Letf :R→R be a 2π-periodic function with f(x) = (x−π)²

4 for x∈[0,2π].

(i) Determine the Fourier coefficients ˆf_k of f.

(ii) Show that

∞

X

k=1

1

k⁴ = π⁴

90 by calculating the right- and left-hand sides of Parseval’s formula

∞

X

k=−∞

|fˆ_k|² = 1 2π

Z π

−π

|f(x)|²dx.