Problem 1. Let x1 , . . . , xn be n positive real numbers. We define
1
p
x1 + . . . + xpn p
for p ∈ R \ {0},
Mp (x1 , . . . , xn ) :=
n
1
M0 (x1 , . . . , xn )
:= (x1 · x2 · . . . · xn ) n
M∞ (x1 , . . . , xn ) := maxi xi
M−∞ (x1 , . . . , xn ) := mini xi .
Prove that the function [−∞, ∞] 3 p 7→ Mp (x1 , . . . , xn ) is continuous and nondecreasing.
Problem 2. Let (V, k · k) be a normed vector space, W ⊂ V a closed linear subspace
and Z = V /W . For z ∈ V let [v] ∈ Z be the equivalence class {v + w : w ∈ W }. Set
k[z]kq := inf kz − wk .
w∈W
Show that k · kq is a norm and that (Z, k · kq ) is a Banach space if (V, k, ·k) is.
Problem 3. Let I ⊂ R be a closed interval and α ∈]0, 1[. Consider in C 0,α (I) the
closure K of C 0,1 (I). Show that u ∈ K if and only if
"
#
|u(x) − u(y)|
lim
sup
= 0.
ε↓0 x,y∈I,0<|x−y|≤ε
|x − y|α
Problem 4. Let f be a nonnegative measurable function on ]0, ∞[ and define
ˆ
1 x
F (x) =
f (t) dt .
x 0
Prove that
(a)
(b)
(c)
(d)
F ∈ L1 if and only if f ≡ 0;
p
kF kLp ≤ p−1
kf kLp for any p ∈]0, +∞[ and any f ∈ Lp ;
Show that the equality in (b) holds if and only if f ≡ 0;
Show that the constant in (b) is optimal.
Problem 5. Let E ⊂ Rn be a measurable set with positive Lebesgue measure. Show
that the set E − E := {x − y : x, y ∈ E} contains a neighborhood of the origin.
Problem 6. Let (X, d) be a metric space and (Y, k · k) a normed vector space. The
map i : X → Y is called an isometric embedding if d(x, y) = ki(x) − i(y)k for every
x, y ∈ X. Show that
(a) if [a, b] ⊂ R is a closed interval and X = C([a, b]), then there a linear isometric
embedding of X into Y = `∞ ;
(b) there is no linear isometric embedding of C([a, b]) into `p if 1 < p < ∞;
1
2
(c) if (X, d) is separable, then there is always an isometric embedding of X in `∞ .
Problem 7. Denote by B the closed unit ball of `∞ . Show that, for any compact convex
symmetric set K ⊂ R2 which contains a neighborhood of the origin, there is a linear
map L : R2 → `∞ such that B ∩ L(R2 ) = L(K).
Problem 8. Show that there is no function f : R → R which is continuous at every
x ∈ Q and discontinuous at every x ∈ R \ Q.
Problem 9. Prove that the strong and weak convergence in `1 are equivalent.