I: (lp(N),k·kp)−→ (l∞(N),k·k∞)
(xn)7→ (xn)
kIkop
(X, k·kX) (Y, k·kY)K
X k ≥0 0 < α ≤1
Lipk,α
0(K, Y ) := {f:K−→ Y:f(0) = 0 kf(x)−f(x0)kY≤kkx−x0kα
X;∀x, x0∈X}.
Lipk,α
0(K, Y ) (C(K, Y ),k·k∞)
K:C([a, b]) →C([a, b]) K(f)(s) = Rb
ak(s, t)f(t)dt
k: [a, b]×[a, b]→R(fn) (C([a, b]),k.k∞)
(1) k
(2) (K(fn))
(3) (K(fn)) X
fn(t) = pt+ 4(nπ)2t∈[0,+∞[
(1)
f= 0
(2) (fn) (C([0,+∞[),k.k∞)
K E f :K→K
∀x, y ∈K
kf(x)−f(y)k≤kx−yk.
a∈K
(1) (fn)
fn(x) = f(1
na+ (1 −1
n)x).
n∈N∗fntn
(2) f
f: [0,1] →RC1f(0) = 1
f0(x) = f(x−x2)
X= (C1([0,1]), N)N(f) = kfk∞+kf0k∞
f T
T(f)(x) = 1 + Zx
0
f(t−t2)dt.
T◦T