Licence 3 Mathématiques

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R2
A:= {(x, y)∈R2|x2−y2−2xy ≤1}
B:= {(x, y)∈R2|x2+y2+exy ≤36}
C:= {(x, y)∈R2|2x2+ 3y2<1}
(E, d) (xn)n≥0E a ∈E
K={xn|n≥0}∪{a}E
(E, d) (xn)n≥0E
a E a
(E, k·k)A B E
A+B:= {a+b|(a, b)∈A×B}.
A A +B
A B A +B
A B A +B
R×R
A:= {(a, b)∈R×R|a > 0, ab = 1}B:= {(a, b)∈R×R|a= 0, b ≤0}.
A, B A +B

E F R¯ BEE u
E F u(¯ BE)F
∀x, y ∈E, u(x+y) = u(x) + u(y).
x∈E r ∈Qu(rx)
M
∀x∈E, ku(x)k ≤ Mkxk.
u
(E, k·k)A E a ∈A
ε > 0B(a, ε)∩A6={a}
f:E→E f(A)⊆A
∀x∈A\ {a},kf(x)−ak<kx−ak.
f(a) = a
x0∈A n ∈Nxn+1 =f(xn) (xn)n∈N
a
n d RnFRn
x0Rn
(fk)k∈Nx0
d(x0, F )+1 d(x0, fk)k∈Nd(x0, F )
d(x0, F )
Rnx7→ d(x, F )U
Rnx∈U d(x, F )
P(x)P:U→F
x∈U(xk)k∈N∈UNx
dxk, P (xk)k∈N
dx, P (x)
d:Rn×Rn→R Rn×Rn
δ(x1, y1),(x2, y2)= max d(x1, y1), d(x2, y2)
P(xk)k∈NP(x)