Codimension

F G KE
F∩G={0}
F= dim G < +∞
F G
F G E F ⊂G
F G G ≤F
F G K
E
F⊂G F =G
F=G
E F G E
F⊂G F E
F G G
E
GF+EG=EF
EKF E G
F G E
EG=EF+FG
ϕ1, . . . , ϕnKE
F=
n
\
i=1
ker ϕi

H F E dim H= dim G
p H F
ker pG= ker p∩G=F∩G={0}pG:G→H
dim H= dim G < +∞pGG H
x∈E a = (pG)−1(p(x)) b=x−a x =a+b a ∈G
p(b) = p(x)−p(a) = p(x)−p(x)=0 b∈ker p=F E =G+F
F F H
K G ∩H H H
G∩K=G∩H∩K={0}K⊂H F ⊂G⊂G+K H ⊂G+K
E=F+H⊂G+K G K K
G G = dim K≤dim H=F K
H
K F E
E=F⊕K F ⊂G
E=G+K
G∩K K K
K0
(G∩K)⊕K0=K
E=G⊕K0
dim K0=G=F= dim K
K=K0
E=G⊕K
G F
x∈G F K E
x=xF+xKxF∈F xK∈K
xK=x−xF∈G∩K
x xFG
xK= 0 x=xF∈F
G⊂F G =F
F E F
H K H ∩G H G
K E K G
E F H ∩G G
F G
dim H= dim K+ dim H∩G
EF=GF+EG
F G G
E F G G
E F E
G H K
H∩F H
F K E K
F E
G H ∩F F G
F
dim H= dim K+ dim H∩F
EG=EF+FG
u:E→Kn
u(x)=(ϕ1(x), . . . , ϕn(x))
u F

u
HKna1, . . . , an∈K
∀x∈E, a1ϕ1(x) + · · · +anϕn(x)=0
(ϕ1, . . . , ϕn) Im u=Rn
u
u F n