FUNCTIONAL ANALYSIS: EXERCISE SHEET 8
DATE OF SUBMISSION: THURSDAY, APRIL 28, 12.00 PM (NOON)
Problem 1. Assume that {un } ⊆ L2 (]0, +∞[) is a sequence such that
Z ∞
u2n (x) dx = 4 for all n.
0
Show that there exists a subsequence {unj } and a function u ∈ L2 (]0, +∞[) such that
Z ∞
Z ∞
unj (x)v(x) dx →
u(x)v(x) dx, as n → ∞,
0
0
2
for all v ∈ L (]0, +∞[).
Problem 2.
• Assume that {un } ⊆ C 0 ([0, 1]) is a sequence such that
|un (x)| ≤ 2, for every x ∈ [0, 1], and for all n ∈ N.
Show that there is a subsequence {unj } and a function u ∈ L∞ (]0, 1[) such that
Z 1
Z 1
unj (x)v(x) dx →
u(x)v(x) dx, for all v ∈ L1 (]0, 1[).
0
0
• Does the function u above necessarily belong to C 0 ([0, 1])?
Problem 3. Let Ω ⊆ RN be an open set, and 1 < p, p∗ < +∞ be such that
Assume kun kLp∗ (Ω) ≤ 1 and
1
p
+
1
p∗
= 1.
∗
• un * u weakly in Lp (Ω), as n → ∞,
• vn → v strongly in Lp (Ω), as n → ∞.
Show that
Z
Z
un (x)vn (x) dx →
u(x)v(x) dx, as n → ∞.
Ω
Ω
Problem 4.
(a) Let T ∈ (R)∗ , that is, T : R → R is a linear functional. Show that
T (x) = λ x, for some λ ∈ R.
(b) Let N > 0 be a fixed natural number and let T ∈ (RN )∗ . Show that
T (x) = y · x, for some y ∈ RN .
In the above expression, y · x denotes the scalar product between x and y. Conclude that
we can identify RN with its dual.
(c) Let {xn } be a sequence in RN and x ∈ RN . Show that the following statements are
equivalent.
(i) xn → x strongly as n → ∞,
(ii) xn * x weakly as n → ∞,
∗
(iii) xn * x weakly star as n → ∞.
Conclude that all the three notions of convergence are equivalent in RN .
1