uk=u|Ik:Ik→Ik+1 Im uk=u(Ik) = u(uk(E)) = uk+1(E) = Ik+1
Ker uk=Ik∩Ker u
dim Ik= rg(uk) + dim Ker uk= dim Ik+1 + dim(Ik∩Ker u)
ak= dim(Ik∩Ker u)≥0Ik+1 ⊂Iku(E)⊂E uk(u(E)) ⊂uk(E)
ak(dim Ik)N
(ak)
d= inf{k∈N/ak= 0}(ak)ak>0
k < d ak= 0 k≥d(dim Ik)d
0
rg uk+ dim Ker uk= dim E= rg uk+1 + dim Ker uk+1,
dim Ik−dim Ik+1 = dim Ker uk+1 −dim Ker ukdim Ker(uk)
d n
dim Ikd0 =
dim Id≤dim I1−d+ 1 d≤rg u+ 1 ≤n
dim Im up+ dim Ker up=n x ∈Im up∩Ker up
y∈E x =up(y) 0 = up(x) = u2p(y)y∈Ker u2p= Ker up
p x =up(y)=0 E= Ker up⊕Im up
u up= 0 u
Xqu E
T u
λ λp= 0 T
(ei)1≤i≤nE T
u(e1) = 0 u(ei)∈Vect(e1· · · , ei−1)i≥2
uk(ei) = 0 i≤k uk(ei)∈Vect(e1,· · · , ei−k)i≥k+ 1
un= 0 u
χu=Xnu