Download ANALISI FUNZIONALE FOGLIO DI ESERCIZI N. 2 Exercise 1. Let (X

ANALISI FUNZIONALE
FOGLIO DI ESERCIZI N. 2
A.A. 2016/17
Exercise 1. Let (X, d) be a metric space. Define
C(X) = {f : X → R : f is continuous}
Cb (X) = {f ∈ C(X) : f is continuous and bounded}
Cc (X) = {f ∈ C(X) : f is continuous and zero outside a compact subset of X}.
For all f ∈ C(X), define the uniform norm of f as
.
kf k∞ = sup |f (x)|.
x∈X
(a) Prove that there are metric spaces X for which (C(X), k · k∞ ) is not
a metric spaces.
(b) Prove that (Cb (X), k · k) is a Banach space.
(c) Prove that there are metric spaces X for which Cc (X) is not closed
in Cb (X).
(d) Define C0 (X) = Cc (X). Find an example of a metric space X such
that C0 (X) \ Cc (X) 6= ∅.
(e) If X is compact, prove that (C(X), k · k∞ ) is a Banach space and
that C(X) = Cb (X) = C0 (X) = Cc (X).
Exercise 2. For x, y ∈ R let
d(x, y) =
|x − y|
.
1 + |x − y|
Prove that d is a distance on R.
Exercise 3. Let a, b ∈ R, a < b. Let α : [a, b] → R be a continuous function
with α(x) ≥ β > 0 for some β > 0. For two functions f, g ∈ Cb ([a, b]), define
dα (f, g) = sup [α(x)|f (x) − g(x)|] .
x∈[a,b]
Prove that dα is a metric. Prove that (Cb ([a, b]), dα ) is a complete metric
space. Is such distance induced by a norm? Which one? Is the eventual
normed space obtained a Banach space? Motivate your answers.
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A.A. 2016/17
Exercise 4. Let Ω ⊂ Rd be an open set. Let {Ωm }m∈N S
be an increasing
sequence of bounded open sets such that Ωm ⊂ Ωm+1 , Ω = +∞
m=1 Ωm . Prove
that the space C(Ω) of continuous functions f : Ω → R is a complete metric
space with the metric
ρ(f, g) =
+∞
X
1 maxx∈Ω |f (x) − g(x)|
.
2m 1 + maxx∈Ω |f (x) − g(x)|
m=1
Prove also that ρ(fm , f ) → 0 if and only if fm converges to f uniformly on
compact subsets of Ω.
Exercise 5. Let 1 ≤ p < q ≤ +∞. Prove that `p ⊂ `q (Hint: use that
P
is +∞
k=1 |yk | < +∞ then yk → 0 as k → +∞). Moreover, prove that if a
sequence {xn }n∈N converges to zero in the k · k`p norm then it converges to
zero in the k · k`q norm.
Exercise 6. Let c = {x ∈ `∞ : limk→+∞ xk exists}. Prove that c is closed
in `∞ . Let c0 = {x ∈ `∞ : limk→+∞ xk = 0} equipped with the `∞ -norm.
Prove that c0 is closed in c.
Exercise 7. Consider the sequence
n
,
for n, k ∈ N.
xn,k = 2
k + n2
For all n ∈ N, let xn := {xn,k }k∈N . Prove that xn ∈ `1 ∩ `∞ for all n ≥ 1.
Prove that xn → 0 in `∞ but xn does not converge to zero in `1 .
Exercise 8. Let (A, d) be a complete metric space, and let Lip(A; R) be
the space of Lipschitz continuous functions from A into R. For every f ∈
Lip(A; R), consider the number
|f (x) − f (y)|
,
d(x, y)
x,y∈A
[f ]Lip := sup
and the quantity
kf kLip = kf k∞ + [f ]Lip ,
where kf k∞ is the usual uniform norm of f , i. e. kf k∞ = supx∈A |f (x)|.
Prove that k · kLip is a norm on Lip(A; R). Prove that (Lip(A; R), k · kLip ) is
a Banach space.
Exercise 9. Let A = [a, b] ⊂ R. Let C(A) the (Banach) space of continuous
functions on A equipped with the usual uniform k · k∞ norm. Prove with an
example that the linear subspace Lip(A) ⊂ C(A) of all Lipschitz continuous
functions on A is not closed in (C(A), k · k∞ ). Explain why this is not a
contradiction with the fact that the space of Lipschitz functions is a Banach
space (and in particular a complete space) from the previous exercise.
ANALISI FUNZIONALE
FOGLIO DI ESERCIZI N. 2
3
Exercise 10. With the notation of the previous exercise, prove that Lip(A)
is dense in (C(A), k · k∞ ) (Hint: for a given continuous function f ∈ C(A),
construct an approximation of f in the uniform norm which is a Lipschitz
function).
Exercise 11. Let A = [a, b] ⊂ R, and let α ∈ (0, 1). Let C 0,α (A) be the
space of α-Hölder continuous functions on A, i. e. the set of functions
f : A → R such that |f (x) − f (y)| ≤ C|x − y|α for all x, y ∈ A. For all
f ∈ C 0,α (A), set
|f (x) − f (y)|
,
|x − y|α
x,y∈A
kf kC 0,α = kf k∞ + [f ]α .
[f ]α := sup
Prove that k · kα is a norm on C 0,α . Moreover, prove that the space C 0,α
equipped with k · kC 0,α is a Banach space.
Exercise 12. Let A = [a, b] ⊂ R, and let α ∈ (0, 1). Let C(A) be the space
of continuous functions on A equipped with the uniform norm. Let
MK := {f ∈ C(A) : kf kC 0,α ≤ K}.
Prove that MK is a compact subset of C(A).
Exercise 13. Let (l2 , d) be the complete metric space given by
(
)
∞
X
l2 = x = {xn }n∈N : xn ∈ R and
x2n < ∞
n=1
and
d(x, y) =
∞
X
!1
2
|xn − yn |2
.
n=1
Show that
(1) The set
Q = {x ∈ l2 : |xn | ≤ 1 for any n ∈ N}
is not bounded.
(2) The set
B = {x ∈ l2 :
∞
X
x2n ≤ 1}
n=1
is bounded but not totally bounded.
(3) The set
H = {x ∈ l2 : |xn | ≤
is compact.
1
for any n ∈ N}
n
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A.A. 2016/17
Exercise 14. Let {fn }n∈N ⊂ C([0, 1]) such that supx∈[0,1] |fn (x)| ≤ 1 for
any n ∈ N. Define Fn : [0, 1] → R by
Z x
fn (t) dt.
Fn (x) =
0
Show that the sequence {Fn }n∈N has a subsequence that converges uniformly
on [0, 1].
Exercise 15. Let G the following subset of C([0, π])
Z π
sin(xy)f (y) dy, f ∈ C([0, π]), kf k∞ ≤ 1 .
G = g ∈ C([0, π]; g(x) =
0
Prove the following statement.
(1) G is relatively compact in in C([0, π]).
(2) G is relatively compact in L1 ([0, π]).
Exercise 16. Let f : R3 → R be a continuous function and let A :
C([0, 1]) → C([0, 1]) the map defined as follows
Z 1
(Au)(x) =
f (x, s, u(s)) ds,
u ∈ C([0, 1]).
0
Prove the following statement.
(1) A is well defined.
(2) A is a continuous map.
(3) If U is a bounded subset of C([0, 1]), A(U) is relatively compact in
C([0, 1]).
Exercise 17. Show that C([0, 1]), the space of continuos function from [0, 1]
R1
to R, with the norm kf k1 = 0 |f (t)| dt is not complete.
Exercise 18. Let X and Y be metric spaces and f : X → Y be a continuous
function. Show that if K ⊂ X is compact then f (K) ⊂ Y is compact as
well.
Exercise 19. Prove that the set
B = {f ∈ C([0, 1]) : kf k∞ ≤ 1}
is not relatively compact with respect to the sup-norm. Hint: find a sequence
in B which admits no convergent subsequences.
Exercise 20. Let f : R×R → R be a continuous function in a neighborhood
of the point (t0 , y0 ) ∈ R2 . Then, there exists ε > 0 such that the ordinary
differential equation
(
y 0 (t) = f (t, y(t))
y(t0 ) = x0
has a solution defined on the interval [t0 − ε, t0 + ε].