Sous-groupes de Rn

Rn
(R,+)
GR
GR
ω > 0G=ωZ:= {ωn, n ∈Z}
f:R→R
Pf:= {T∈R|f(x+T) = f(x)∀x∈R}.
(Pf,+) (R,+)
fPf6={0}
f
Pf=R;Pf={0};Pf= 2πZ;Pf=Z;Pf=Q.
h:R→R
ω > 0Ph=ωZ
ω1>0ω2>0q:= ω1
ω2
h h :R→R
∀x∈R, h(x) = h(x+ω1) = h(x+ω2).
q6∈ Qh
h1h2Ph1=ω1ZPh2=ω2Z
h1+h2q∈Q
T h1+h2x7→ h1(x+T)−
h1(x)
q6∈ Q
sup
R
(h1+h2) = sup
R
h1+ sup
R
h2.
hi(xi) = sup hi(i= 1,2) pnqn
pnω1+qnω2→
n→∞
x1−x2.
(1) (Rn,+)
GRnG
G
G
G6={0}g6= 0 G H := G∩Rg
R
G
R

G/H Rn/Rg
RnB(0,kgk)∩G
G