Esercizi svolti di analisi reale e complessa - Dipartimento di ...

ℓ 1 ℓ ∞
x = {xn}n ∈ R N
x1 ≡
n∈N
R N
|xn| e x∞ ≡ sup |xn| .
n∈N
ℓ 1 ≡ {x ∈ R N : x1 < ∞}
ℓ ∞ ≡ {x ∈ R N : x∞ < ∞} .
1 (ℓ 1 , · 1) (ℓ ∞ , · ∞)
2 ℓ 1 ⊂ ℓ ∞ (ℓ ∞ , · ∞)
(ℓ 1 , · ∞)
3 ∗∗ ℓ 1 · ∞
(ℓ 1 , · 1)
Ω := {x ∈ ℓ 1 : x1 ≤ 1}
D := {x ∈ ℓ 1 : |xk| ≤ 1 ∀ k, xk = 0 ∀ k > 10} .
R n R m
. . .
f(x) = xi
x
ɛ > 0 δ |f(x) − f(x0)| < ɛ
|x − x0| < δ
f = |x| α , α > 0 x ∈ R 4 , x0 = (0, 1, 1, 2)
x0 = (0, .., 0)
f = sin 1
x1x2 , x ∈ R
3
3 , x0 = (−1, 0, −1)