REAL ANALYSIS: HOMEWORK SET 7
DUE FRI. DEC. 7
Pn
Exercise 1. Let φn ∈ (`∞ )∗ be given by φn (f ) = n1 j=1 f (j) (for f ∈ `∞ ).
Using Alaoglu’s theorem, stating that for any normed vector space X the set {ϕ ∈
X ∗ | kϕk ≤ 1} is compact in the weak∗ topology, show that there is a subsequence
{φnk } converging weak∗ to some φ ∈ (`∞ )∗ and that the limit φ does not arise from
an element of `1 .
Exercise 2. Let {fn } ⊆ Lp with kfn kp uniformly bounded and fn → f a.e.
(1) For 1 < p < ∞ show that fn → f weakly in Lp
(2) For p = 1, find counter examples in L1 (R, m) and in `1 .
(3) For p = ∞, µ σ-finite, show that fn → f in weak∗ topology of (L1 )∗ = L∞ .
Exercise 3. Let 1 < p < ∞, and let {fn } ⊆ Lp .
(1) Assume that kfn kp is uniformly bounded and show that fn → 0 weakly iff
R
fn g → 0 for all simple g ∈ Lq .
(2) Give an example in Lp (R) showing that this is not true without the uniform
bound on the norms.
(3) Show that in `p we have that fn → f weakly if and only if kfn kp is uniformly
bounded and fn → f pointwise.
Exercise
4. Let K be a nonnegative measurable function on (0, ∞) with φ(s) =
R∞
s−1
K(x)x
dx < ∞ for all 0 < s < 1.
0
(1) Let p, q ∈ (0, ∞) with p−1 + q −1 = 1 and f, g ∈ L+ (0, ∞) and show that
Z ∞
1/p Z
1/q
Z ∞Z ∞
−1
p−2
p
q
K(xy)f (x)g(y)dxdy ≤ φ(p )
x f (x) dx
g(x) dx
.
0
0
0
R
(2) Show that the operator T f (x) = K(xy)f (y)dy is bounded on L2 (0, ∞)
with kT k ≤ φ( 21 ).
(3) Compute T and φ for the special case where K(x) = e−x .
Exercise 5. Let µ be a Radon measure on X (a locally compact Hausdorff space).
We define the support of µ the complement N c where N is the union ofR all open
sets U in X with µ(U ) = 0. Show that µ(N ) = 0 and that x ∈ suppµ iff f dµ > 0
for every f ∈ Cc (X) with values in [0, 1] and f (x) > 0.
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