Algèbre et géométrie II Série 2

X
d:X×X−→ R,(x, y)7−→ (1x6=y
0
d X
X Y fn:X→Y
f:X→Y ε > 0
Nε∈Nx∈X n ≥Nεd(fn(x), f(x)) < ε
fnf
X Y f :X→Y
f
x∈X(xn)n∈N⊂Xlimn→∞ xn=x
limn→∞ f(xn) = f(x)
U∀u∈U, ∃ε > 0Bε(u)⊆U.
V⊂Y f−1(V)
(xn) (xn)6⊂ f−1(V) limn→∞ xn=x x ∈f−1(V)
V
Lp
p∈[1,∞[
C0
c(R,R) := {f:R→R|f}.
f f

C0
c(R,R)f∈C0 c(R,R)
kfkp=Z∞
−∞
|f(x)|pdx1/p .
L2L1
V=C0
c(R,R)L1L2(fn)n∈N⊂V
L2L1
X d A ⊂X x ∈X
d(x, A) := inf{d(x, a)|a∈A}.
X→R, x 7→ d(x, A)
d(x, A)=0 x∈A
A A
A
A=\{F|A⊆F, F }.