Let (X n , d n ) be a family of metric spaces and X := Y

SS 2021 M. Röckner

Exercises for Functional Analysis

Exercise 1 Submission date: Friday, 23.04.2021 Digital submission via the E-Learning site of the tutorial Exercise 1.

Let (X _n , d _n ) be a family of metric spaces and X := Y

n∈ N

X n = {(x _n ) n∈ N | x n ∈ X n für n ∈ N } the cartesian product of the sets X _n , n ∈ N.

a) Set

d: X × X → R , (x, y) 7→

∞

X

n=1

2 ⁻ⁿ d _n (x _n , y _n ) 1 + d n (x n , y n ) .

Prove that (X, d) is a metric space. (2 Points)

b) Prove that (X, d) is complete if and only if (X _n , d _n ) is complete for all n ∈ N.

(2 Points) Exercise 2.

The space ` ¹

R is dened by:

` ¹ _R := {(x _n ) n∈ N |

∞

X

n=1

|x _n | < ∞}.

For x = (x n ) n∈ N ∈ ` ¹ _R set

kxk := sup

n∈ N

n

X

k=1

x k

. Prove that (` ¹

R , k · k) is a normed space. (2 Points)

Is (` ¹ _R , k · k) a Banch space? Prove it or construct a counter-example (2 Points) Exercise 3.

The spaces ` ^p

R are dened by:

` ^p

R := {(x _n ) n∈ N |

∞

X

n=1

|x _n | ^p < ∞}, p ∈ [1, ∞) and

` ^∞ _R := {(x _n ) n∈ N | sup

n∈ N

|x _n | < ∞}.

For which s ∈ R and p ∈ [1, ∞] does (n ^s ) n∈ N ∈ ` ^p

R hold? (4 Points)

1

Exercise 4.

Let (X, B, µ) be a mesure space with a nite measure µ . Let (Y, d) be a metric space. We set:

M (B, d) := {f : X → Y | f is B/B(Y )-measurable } and

D _µ : M (B, d) × M(B, d) → R , D _µ (f, g) :=

Z d(f, g) 1 + d(f, g) dµ.

a) Prove that D µ is a pseudometric on M(B, d) .

(1 Point) b) Prove that the sequence (f n ) n∈ N in M (B, d) converges in measure µ to a f ∈ M (B, d) (i.e. µ(d(f, f n ) >

ε) → 0 ) if and only if lim n→∞ D µ (f, f n ) = 0 holds.

Hint: Consider

ε

1 + ε µ(d(f _n , f ) > ε) = Z

{d(f

,f)>ε}

ε 1 + ε dµ and use the fact that the mapping x 7→ _1+x ^x is increasing.

(2 Points) c) Consider the equivalence relation

f ∼ g :⇔ f = g µ-almost everywhere.

Then, ((M (B, d)/ ∼, D _µ ) is a metric space (No proof necessary!). Show that under the additional ass- umption (Y, d) being complete, that ((M (B, d)/ ∼, D _µ ) is complete as well.

(1 Point)

2