Suites et Séries de fonctions.

(un)n∈NPn∈Nun
unn
(fn)n∈N
Pn∈Nfn
x
x fn(x)
f(x).
fn:I→RI
Rfnf I
quelquesoit xdans I, fn(x)→n→∞ f(x).
I= [0,1] fn(x) = 
nx si x∈[0,1
n]
−nx + 2 si x∈[1
n,2 n]
0 si x∈[2
n,1]
x= 0 ,
fn(x) = 0 →0n→ ∞ x6= 0 n>E(2
x) + 1 fn(x)=0
fn(x)→0n→ ∞ fnf
f(x) = 0 x∈[0,1] n xn=1
n
fn(xn) = 1
1 = supx∈[0,1] |fn(x)|0 = supx∈[0,1] |f(x)|
limn→∞ supx∈[0,1]
E E
d:E×E→R+
(D1) ∀(x, y)∈E2,[d(x, y) = 0 ⇐⇒ x=y]
(D2) ∀(x, y)∈E2, d(x, y) = d(y, x)
(D3) ∀(x, y, z)∈E3, d(x, y)≤d(x, z) + d(z, y).
(E, d)

E+
R C
N E
N:E→R+
(N1) ∀X∈E , [N(X) = 0 ⇐⇒ X= 0]
(N2) ∀X∈E , N(λX) = |λ|N(X)
(N3) ∀(X, Y )∈E2, N(X+Y)≤N(X) + N(Y)
(E, N )N(f)
N(f) = kfk
(E, kk)
d(X, Y ) = kX−YkE
kk
d(X, Y ) = 0 kX−Yk= 0 (N1) X=Y.
d(X, Y ) = kX−Yk=k(−1)(Y−X)k=| − 1|.kY−Xk=d(Y, X)
d(X, Y ) = kX−Yk=kX−Z+Z−Yk ≤kZ−Xk+kZ−Yk=d(X, Z) +
d(Z, Y )
Rnk(x1, x2, . . . , xn)k2=px2
1+x2 2+. . . +x2
nk(x1, x2, . . . , xn)k∞=
max(|x1|,|x2|, . . . |xn|)k(x1, x2, . . . , xn)k1=|x1|+|x2|+. . . +|xn|
kk2,
. . .
Cnk(z1, z2, . . . , zn)k2=√z1z1+z2z2+. . . +znznk(z1, z2, . . . , zn)k∞=
max(|z1|,|z2|, . . . |zn|)k(z1, z2, . . . , zn)k1=|z1|+|z2|+. . . +|zn|
(E, kk)f
r B(f, r) = {g∈E, kf−gk< r}
f r B(f, r) = {g∈E, kf−gk ≤ r}
R2
kk2(x0, y0)
rkk∞(0,0) 1
(1,1),(−1,1),(−1,−1) (1,−1) kk1(0,0)
1 (1,0),(0,1),(−1,0) (0,−1)
N1N2
α β
∀x∈E , αN1(x)≤N2(x)≤βN1(x).

E
α β R2kk2et kk1
(E, kk)
fn→fkk kfn−fk →n→∞ 0
E I R
kfk∞= sup
x∈I|f(x)|
fnf I
kfn−fk → 0n→ ∞
E[a, b]R
kfk1=Zb
a|f(t)|dt
fnfkfn−fk1→n→∞
0
E[a, b]R
kfk2=Zb
a|f(t)|2dt
fnf
kfn−fk2→n→∞ 0

fnf ε > 0n0
n≥n0fnf
ε ε f
fn2ε
f
fn
[0,1
n2]n x = 0 0 x=1 n21≥x≥1 n2
fn(0) = n
f(0).
fnf I fn
f I
x0|f(x0)−fn(x0)| ≤ supx∈I|fn(x)−f(x)|=kfn−fk∞→n→∞
0|fn(x0)−f(x0)| → 0fn(x0)→f(x0)fn
f n → ∞.¤
f
supx∈I|fn(x)−
f(x)|
I= [0,∞[ et fn(x) = nx
1+n2x. x = 0 fn(0) = 0 →0n→ ∞
x6= 0, fn(x)→0n→ ∞ fn
f(x) = ½0 si x= 0
0 si x6= 0 = 0 fnf0(x) =
1
(1+n2x)2>0fn(x)→1 nx→ ∞ fn
0x= 0 1
n
kfn−fk∞=1
n→n→∞ 0fn
I n → ∞
I= [0,∞[ et fn(x) = nx
1+nx2. x = 0 fn(0) = 0 →n→∞ 0
x6= 0 fn(x)→n→∞ 0fnf= 0 n→ ∞
f0
n(x)=1−n2x
(1+nx2)2xn=1
√n

supx∈I|fn(x)−f(x)|x=1
√n
fn(1
√n) = 1
2√n→n→∞ 0fn
I n → ∞
I= [0,∞[ et fn(x) = nx
1+nx . x = 0 fn(0) = 0 →n→∞ 0x6=
0fnf(x) = ½0 si x= 0
1 si x6= 0 = 0
x6= 0 |fn(x)−1|=|nx
1+nx −1|=|1 1+nx |1
x→0+kfn−fk∞= 1 6→ 0n→ ∞ fn
f n → ∞.
fn→f I R
n∈Ngnx0∈I f
x0
|f(x)−f(x0)| ≤ |f(x)−fn(x)|+|fn(x)−fn(x0)|+|fn(x0)−f(x0)|
ε > 0,||fn−f||∞→0n→ ∞ N n ≥N
||fn−f||∞≤ε
3n=N fNx0η > 0
|x−x0|< η |fN(x)−fN(x0)| ≤ ε
3n≥N
|f(x)−f(x0)| ≤ |f(x)−fn(x)|+|fn(x)−fn(x0)|+|fn(x0)−f(x0)|n=N
|f(x)−f(x0)| ≤ ε∀ε > 0,∃η > 0,tel que si |x−x0|<
ηalors |f(x)−f(x0)|< ε f x0¤
lim
n→∞ lim
x→0+fn(x)6= lim x→0+lim
n→∞ fn(x)
fn(x) = nx
nx+1 [0,∞[f(x) =
½1 si x > 0
0 si x = 0 0 = lim
n→∞ lim
x→0+fn(x)6= lim x→0+lim
n→∞ fn(x) = 1
fn:I→RSn=
n
X
k=0
fk.
PfnS I Sn
S I PfnS I
SnS I
PfnI
Panan
nsup
x∈I|fn(x)|=kfnk∞≤anPan<∞)
PfnI,
I
x n
P|fn(x)|nPan