Question
Transcribed Text
3. Construct a sequence of integrable functions {fk} on [0, 2π] such that
lim
k→∞
1
2π
Z 2π
0
|fk(θ)|
2
dθ = 0
but limk→∞ fk(θ) fails to exist for any θ.
[Hint: Choose a sequence of intervals Ik ⊂ [0, 2π] whose lengths tend to 0, and
so that each point belongs to infinitely many of them; then let fk = χIk
.]
5. Let
f(θ) = (
0 for θ = 0
log(1/θ) for 0 < θ ≤ 2π,
and define a sequence of functions in R by
fn(θ) = (
0 for 0 ≤ θ ≤ 1/n
f(θ) for 1/n < θ ≤ 2π.
Prove that {fn}∞
n=1 is a Cauchy sequence in R. However, f does not belong to
R.
[Hint: Show that R b
a
(log θ)
2 dθ → 0 if 0 < a < b and b → 0, by using the fact that
the derivative of θ(log θ)
2 − 2θ log θ + 2θ is equal to (log θ)
2
.]
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