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E
n∈N∗
u∈ O(E)
Sp(u)⊂ {−1,1}
u
n∈N∗
<, >: (A, B)∈ Mn(R)27→ tr(tAB)
A∈ Mn(R)
M7→ AM
f:E→E
f(0) = 0 ∀x, y ∈E, kf(x)−f(y)k=kx−yk
f
∀x∈E, f(−x) = −f(x)
f
B= (ei)1≤i≤nE
∀x∈E, f(x) =
n
X
i=1
< ei, x > f(ei)
f
u∈ O(E)
Ker(u−IdE) = (Im(u−IdE))⊥
n∈N∗
un=1
n
n−1
X
k=1
uk
x∈E(un(x))
x Ker(u−IdE)
E f ∈ L(E)
f∈ O(E)
f2=−
f(x)x x ∈E
SO(E)O(E)
ESOn(R)
u∈ L(E)u∈ SO(E)u
E(x, y, z, t)∈E4
x∧(y∧z) =< x, z > y−< x, y > z
(x∧y)∧z=< x, z > y−< y, z > x
< x, y >2+kx∧yk2=kxk2kyk2

x, y z
x∧(y∧z)=(x∧y)∧z?
< x ∧y, z ∧t >=< x, z >< y, t > −< x, t >< y, z >
E(a, b)∈E2
a6= 0
a∧x=b
E
By∈E ϕyE
∀x∈E, ϕy(x) = x∧y
ϕyB
ϕ3
y=−kyk2ϕy
eϕy(x)x, y ∈E y 6= 0
ϕyE
eϕy=
+∞
X
n=0
1
n!ϕn
y
fR3
A=1
4
3 1 √6
1 3 −√6
−√6√6 2
A=1
9
−8 4 1
474
1 4 −8
E
B= (ei)1≤i≤3f∈ L(E)
B
A=1
2
1−√2 1
√2 0 −√2
1√2 1
.
B0= (u, v, w)E
v, w ∈ P :x+z= 0
fB0f
R3
2x−y+ 3z= 0
D:x=y=zπ
4
R3
D:x=y=zD0:x=y=−z