Scribd
Upload
27 views1 page

Es Convergence 2

The document outlines exercises related to convergence in Lp spaces for a PhD course in Statistics taught by Annalisa Cesaroni. It includes problems on strong and weak convergence, demonstrating various properties and relationships between functions in Lp spaces. The exercises require proofs regarding limits and convergence behaviors under different conditions.

Uploaded by

WILMOIS MASTER
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
27 views1 page

Es Convergence 2

The document outlines exercises related to convergence in Lp spaces for a PhD course in Statistics taught by Annalisa Cesaroni. It includes problems on strong and weak convergence, demonstrating various properties and relationships between functions in Lp spaces. The exercises require proofs regarding limits and convergence behaviors under different conditions.

Uploaded by

WILMOIS MASTER
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 1

Functional Analysis- A.A.

2018-2019
PhD in Statistics
teacher Annalisa Cesaroni
Some other problems on convergence in Lp spaces.
Exercise 1. Let Ω ⊆ Rn be a open set, fk , f ∈ Lp (Ω), for all k ∈ N.
(1) Show that if fk → f (strongly) in Lp (Ω) then limk kfk kp = kf kp .
(2) Show that if fk * f (weakly) in Lp (Ω) then lim inf k kfk kp ≥ kf kp .
p
Exercise 2. Let Ω ⊆ Rn be a open set, let p ≥ 1 and q = p−1 .
p q
Let fk , f ∈ L (Ω), for all k ∈ N, gk , g ∈ L (Ω), for all k ∈ N.
(1) Assume fk → f (strongly) in Lp (Ω) and gk → g (strongly) in Lq (Ω).
Show that fk gk → f g (strongly) in L1 (Ω).
(2) Assume fk → f (strongly) in Lp (Ω) and gk * g (weakly) in Lq (Ω) (this implies
in particular that kgk kq ≤ M ).
Show that fk gk → f g (strongly) in L1 (Ω).
Exercise 3. Let Ω ⊆ Rn be a open set, and fk , f ∈ L2 (Ω), for all k ∈ N such that
fk * f weakly in L2 (Ω).
Show that if limk kfk k2 = kf k2 , then fk → f strongly in L2 (Ω).
1
Exercise 4. Let Ω = (0, +∞) ⊆ R, p > 1 and consider fk (x) = k p e−kx .
(1) Show that fk → 0 almost everywhere.
(2) Show that fk 6→ 0 strongly in Lp (0, +∞).
(3) Show that fk * 0 weakly in Lp (0, +∞).
(4) Show that fk → 0 strongly in Lq (0, +∞) for all 1 ≤ q < p.
Exercise 5. Let Ω = (0, +∞) ⊆ R, and consider fk (x) = χ[k,k+1] (x).
(1) Show that fk → 0 almost everywhere.
(2) Show that fk 6→ 0 strongly in Lp (0, +∞) for any p ∈ [1, +∞).
(3) Show that fk * 0 weakly in Lp (0, +∞) for any p > 1.
(4) Show that fk 6* 0 weakly in L1 (0, +∞).
n
Exercise 6. Let p > 1 and g ∈ ∩1≤q≤p Lq (Rn ). We consider gk (x) = k p g(kx).
(1) Show that gk * 0 weakly in Lp (Rn ).
(2) Show that gk 6→ 0 strongly in Lp (Rn ).
(3) Show that gk → 0 strongly in Lq (Rn ) for all 1 ≤ q < p.

You might also like