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

4th Tutorial Analysis II (engl.)

Summer Semester 2010

(T4.1)

1. Let (X, d) be a metric space, and let x∈X and >0. Recall that U(x) ={y∈X :d(x, y)< }.

For the purposes of this exercise we define

K(x) :={y∈X :d(x, y)≤}.

Give an example where U(x)6=K(x).

Hint: Consider X = (R\[¹₂,³₂])∪ {1} with the metric inherited from the natural metric on R.

2. Let n ∈N and let (X₁, d₁), . . . ,(X_n, d_n) be nonempty metric spaces.

i. Consider the product space X = Πⁿ_k=1X_k with metric d : X×X →R defined by

d((x₁, . . . , x_n),(y₁, . . . , y_n)) = max

1≤k≤nd_k(x_k, y_k).

Prove that this is really a metric onX.

ii. For 1≤k ≤n we let the kth coordinate projection π_k:X →X_k be defined by π_k(x₁, . . . , x_n) =x_k. Prove that coordinate projections are continuous.

(T4.2)

1. LetXbe a metric space,Y ⊂X a subset regarded as a metric space with the induced metric. Show that a subsetV ⊂Y is open relative toY if and only ifV =U∩Y for some open set U ⊂X.

2. Let X be a metric space, and suppose that K and Y are subspaces of X with K ⊂ Y ⊂ X. Prove that K is compact relative to X if and only if K is compact relative to Y. Conclude that the property that K is compact is independent of the space in which K is embedded, and is anintrinsic property of K.

(T4.3) Let a < b ∈ R and consider the set C¹[a, b] of continuously differentiable functions f : [a, b]→R. Prove that C¹[−1,1] equipped with the supremum norm k · k∞ is not a Banach space. (Hint: Consider the sequence (f_n)n∈N given by f_n(x) = p

x²+ 1/n.

Show that this sequence does not converge inC¹[a, b].)