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R N
N∗N
nMn(R)
nR
AMn(R) (A)A
EE
E
Im ` ` E Ker `
k∈N`kE `0=Ek= 0
`k=`◦`k−1
BEMatB(`)
`B
F
dim F F
Vect(u, v) Vect(u, F )
u v
u F
Ekuk
u
E(u|v)
u v O(E)E
E F ⊥
F `∗`
` x E n N∗
Ln(x) = 1
nPn−1
k=0 `k(x). E `
Ln(x)n∈N∗n+∞
Ln(x)n∈N∗
E
B= (e1, e2, e3, e4)
s E
MatB(s) = 
−1
302 3
2
3
0−1
3−2 3
2
3
2
3−2 3
1
30
2
3
2
301 3
.
s
s
S2.
s E 1
−1
E1E−1s
1−1
E1E−1
E
s E1
E−1
E u1=e1+e3+e4
u2=e1+e2+ 2e4u3=−e1+e2+e3.

s(u1)s(u2).(u1, u2)
E1.
E1.
u4=ae1+be2+ce3+de4
u1u2u3.
(u3, u4)E−1.
x E n N∗Sn(x) = 1
nPn−1
k=0 sk(x).
x∈E x =y+z y ∈E1z∈E−1
k∈Nαksk(x) = y+αkz.
n∈N∗βnSn(x) = y+βnz
Sn(x)n∈N∗E
n+∞
x s(x)
` E
MatB(`) = 
3
401 40
03
401 4
1
403 40
01
403 4
.
u=ae1+be2+ce3+de4E
kuk2− k`(u)k2.
k`(u)k6kuk.
u
k`(u)k=kuk.
`
2.
`
`
` λ 6= 1
G1Gλ`
1λ E
x E n ∈N∗Ln(x) = 1
nPn−1
k=0 `k(x).
x∈E x =y+z y ∈G1z∈Gλ
k∈N`k(x)y z k
nN∗Ln(x)y z n
Ln(x)n∈N∗E
n+∞
t E
MatB(t) = T=
1
√301 √3
1
√3
01
√3−1 √3
1
√3
−1
√3
1
√3
1
√30
−1
√3−1 √301 √3
.
T
E ε1=1
√2(e3+e4)
ε2=1
√2(e3−e4).
F1= Vect(e1, ε1). t(e1)
t(ε1)
F1E
t
F2=F⊥
iF1. F2
t
(e2, ε2)F2.
B0= (e1, ε1, e2, ε2)E
T0= MatB0(t).
T0T0

θ= Arcsin q2
3. T 0
θ
F1(e1, ε1)
F2(e2, ε2)
F1
F2t
k∈Nθ k tk
B0
ω∈Rn∈N∗. ζn(ω) = Pn−1
k=0 eikω .
ζn(ω)ω.
ωζn(ω)n∈N∗
x E n ∈N∗Tn(x) = 1
nPn−1
k=0 tk(x).
F1Tn.
y=αe1+βε1∈F1.
tk(y) = γke1+δkε1Tn(y) = λne1+µnε1.
Vk∈ M2(R)
γk
δk=Vkα β.
Un∈ M2(R)
λn
µn=Unα β.
Vkθ k Un
θ n
Tn(y)n∈N∗E
n+∞
x∈E x =y+z y ∈F1z∈F2
Tn(x)n∈N∗n+∞
E
` E x E n N∗
Ln(x) = 1
nPn−1
k=0 `k(x).
E
`∈O(E)
Ker(`−E) Im(`−E)
E
x∈E y ∈Ker(`−E)
z∈E x =y+`(z)−z.
k∈N`k(x)y z k
Ln(x)y z n
Ln(x)n∈N∗
n+∞
E
B(E)h E
x E kh(x)k6kxk.
E
φ∈ L(E)φ∗E
∀x, y ∈E, (φ(x)|y)=(x|φ∗(y))
Id∗
E=IdEλ∈R
φ1, φ2∈ L(E),(λφ1+φ2)∗=λφ∗
1+φ∗ 2
f∈B(E)
f∗B(E)
x∈E f(x) = xkf∗(x)−xk260.
Ker(f−E) = Ker(f∗−E)
ϕ E rg(ϕ∗) = rg(ϕ))

Ker(f−E) Im(f−E)
E E
ϕ E
Ker(ϕ∗) = Im ϕ⊥
E
`∈B(E)x E n N∗
Ln(x)
x
Ker(`−E)∩Im(`−E). y E x =`(y)−y.
n∈N∗`n(y)x y n
Ker(`−E) Im(`−E)
E
x∈ELn(x)n∈N∗E
n+∞
E ` ∈O(E)e
E x ∈E σe(x) = x−2(x|e)
kek2e.
σe(e). x e σe(x). σe
E
σeVect(e)⊥
W= Ker(`−E)W E
u E u /∈W
e=`(u)−u.
e W
σe`(u)−uσe`(u) + uσe`(u)
σe(u)
Vect(u, W ) = Ker(σe◦`−E).
` p
p k = dim W n = dim E