FUNCTIONAL ANALYSIS LECTURE NOTES CHAPTER 3. BANACH ...

2 CHRISTOPHER HEIL
Exercise 1.4. Show that if 1 ≤ p ≤ ∞ then · p,w defines a semi-norm on ℓp w, and it is a
norm if w(i) > 0 for all i.
In particular, if I = {1, . . . , n} then ℓp w = Fn , and each choice of p and w gives a semi-norm
or norm on Fn .
Hints: The Triangle Inequality on ℓp is often called Minkowski’s Inequality. It is easy to
prove if p = 1 or p = ∞. There are several ways to prove it for other p. One ways is to begin
with
= |xi + yi| p =
|xi + yi| p−1 |xi + yi|
x + y p p
i∈I
i∈I
≤
|xi + yi| p−1 |xi| +
|xi + yi| p−1 |yi|.
i∈I
Then apply Hölder’s Inequality to each sum using the exponent p ′ on the first factor and p
for the second (recall that p ′ = p/(p − 1)). Then divide both sides by x + y p−1
p .
Definition 1.5. Let (X, Ω, µ) be a measure space. Given a measurable f : X → [−∞, ∞]
(if F = R) or f : X → C (if F = C), set
⎧
⎪⎨ |f(x)|
fp = X
⎪⎩
p 1/p dµ(x) , 0 < p < ∞,
ess sup |f(x)|,
x∈X
p = ∞,
where these quantities could be infinite. Define
L p
(X) = f : X → [−∞, ∞] or C : fp < ∞ .
Other notations for L p (X) are L p (µ), L p (X, µ), L p (dµ), L p (X, dµ), etc.
When we write L p (R n ), it will be assumed that µ is Lebesgue measure on R n , unless
specifically stated otherwise. In this case we will write dx instead of dµ(x).
The space ℓ p w (I) is a special case of Lp (X), where X = I and µ is a weighted counting
measure on I.
Exercise 1.6. Show that if 1 ≤ p ≤ ∞ then · p is a semi-norm on L p (X), and it is a norm
if we identify functions that are equal almost everywhere.
The Triangle Inequality on L p is often called Minkowski’s Inequality, and its proof is similar
to the proof of Minkowski’s Inequality for ℓ p .
Exercise 1.7. Show that every subspace of a normed space is itself a normed space (using
the same norm).
Definition 1.8 (Distance). Let · be a norm on X. Then the distance from x to y in X
is d(x, y) = x − y.
i∈I