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(0,0)
f(x, y) = x3
y
f(x, y) = x+2y
x2−y2
f(x, y) = x2+y2
|x|+|y|
(0,0)
f(x, y) = x3+y3
x2+y2
f(x, y) = xy
x4+y4
f(x, y) = x2y
x4+y2
f(x, y) = xy
x−y
(0,0)
f(x, y) = sin xy
√x2+y2
f(x, y) = 1−cos(xy)
xy2
f(x, y) = xy= eyln x
f(x, y) = sh xsh y
x+y
f:R+×R∗
+→Rf(x, y) = xyx > 0f(0, y)=0
f
R+×R+
f:R→RC1F:R2\ {(0,0)} → R
F(x, y) = f(x2+y2)−f(0)
x2+y2
lim(x,y)→(0,0) F(x, y)
f:R2→R
f(x, y) = (1
2x2+y2−1x2+y2>1
−1
2x2
f
AR2f:A→R
a b A y f(a)≤y≤f(b)
x∈A f(x) = y
g:R2→RCO R > 0
A B C
g(A) = g(B)
C D C
g(C) = g(D)
E1E2E
E=E1∪E2
f:E→F
f1f2E1E2

AR2xR2
d(x, A) = inf
a∈Akx−ak
d:R2→R
E[a;b]
a
N:E→Rf∈E
N(f) = infk∈R+∀(x, y)∈[a;b]2,f(x)−f(y)≤k|x−y|
E
A(E, N )L
A E
L
f∈ L
Kf=k∈R+| ∀(x, y)∈A2, N(f(x)−f(y)) ≤kN(x−y)
c(f) = inf Kfc(f)∈Kf
a∈A Na:L → R+
Na(f) = c(f) + N(f(a))
NaL
a, b ∈A NaNb
E T :E→E
T(u) = (ukuk ≤ 1
u
kuk
T
E
f(x) = 1
max(1,kxk)x
f
E f
E F f ∈ L(E, F )
(un) 0Ef(un)
f
N1N2EIdE
(E, N1) (E, N2)
d(fn)R R
d
d
E N
p q KE
∀x∈E, N ((p−q)(x)) < N (x)
p q
E=Mp(C)
kMk= max
1≤i,j≤p|mi,j |

XCpPGLp(C)
φ(M) = MX ψ(M) = P−1MP
f(M, N ) = M N
A∈ Mp(C) (kAnk)
A
B∈ Mp(C) (Bn)C
C2=C C
B
a= (an)∈`∞(R)u= (un)∈`1(R)
ha, ui=
+∞
X
n=0
anun
ha, uiϕu:a7→ ha, ui
ψa:u7→ ha, ui
E=`∞(R)
N∞u= (un)∈`∞(R)T(u) ∆(u)
T(u)n=un+1 ∆(u)n=un+1 −un
T∆E
E=C([0 ; 1],R)k.k∞
kfk∞= sup
[0;1] |f|
ϕ:f7→ f(1) −f(0)
E=C0([0 ; 1],R)F=C1([0 ; 1],R)N1N2
N1(f) = kfk∞N2(f) = kfk∞+kf0k∞
T:E→F f : [0 ; 1] →RT(f): [0 ; 1] →R
T(f)(x) = Zx
0
f(t) dt
T
E=C([0 ; 1],R)k.k2f ϕ E
Tϕ(f) = Z1
0
f(t)ϕ(t) dt
Tϕ
E=C([0 ; 1],R)k.k1
kfk1=Z1
0|f(t)|dt
ϕ:f7→ Z1
0
tf(t) dt
R[X]N1N2
N1(P) =
+∞
X
k=0 P(k)(0)N2(P) = sup
t∈[−1,1] |P(t)|
N1N2R[X]
N1
N2

N1N2
E=C([0 ; 1],R)k.k∞
u:f7→ u(f)u(f)(x) = f(0) + x(f(1) −f(0))
E
a= (an)∈`∞(R)u= (un)∈`1(R)
ha, ui=
+∞
X
n=0
anun
ha, uiϕu:a7→ ha, ui
ψa:u7→ ha, ui
K∈ C [0 ; 1]2,R∀(x, y)∈[0 ; 1]2, K(x, y) = K(y, x)
E=C([0 ; 1],R)f∈E
Φ(f): x∈[0 ; 1] →Z1
0
K(x, y)f(y) dy∈R
Φ∈ L(E)
Φkk∞kk1
∀f, g ∈E, (Φ(f)|g)=(f|Φ(g))
Ω = max
0≤x≤1Z1
0|K(x, y)|dy−1
∀λ∈]−Ω ; Ω[,∀h∈E, ∃!f∈E, h =f−λΦ(f)
λ∈R∗
dim ker(Φ −λId) ≤1
λ2ZZ[0;1]2
K(x, y)2dxdy
A B E
A∪B A ∩B A B
E n ≥2
S={x∈E| kxk= 1}
E n ≥2
H E E \H
F p ≤n−2E\F
GLn(R)
GLn(C)

SO2(R)
f:I→Rf
ϕ(x, y) = f(x)−f(y)
X=(x, y)∈I2, x < y
f:I→Rf0
f0
A=(x, y)∈I2, x < yI2
δ:A→Rδ(x, y) = f(y)−f(x)
0∈δ(A)
N Mn(R)N