19 Espaces vectoriels euclidiens. Groupe orthogonal

E n ≥1.
φ:E×E→R
φ(x, x)≥0x E
x E φ (x, x) = 0 x= 0.
E
E n ≥1.
(x, y)7−→ ⟨x|y⟩
E y =x,
∥x∥=⟨x|x⟩
EL(E)E
∀u∈ L(E),∥u∥= sup
x∈E\{0}
∥u(x)∥
∥x∥

x, y E
⟨x|y⟩=1
2∥x+y∥2− ∥x∥2− ∥y∥2
=1
4∥x+y∥2− ∥x−y∥2
∥x+y∥2+∥x−y∥2= 2 ∥x∥2+∥y∥2
x, y E
|⟨x|y⟩| ≤ ∥x∥∥y∥,
x y
x, y E
∥x+y∥ ≤ ∥x∥+∥y∥
x= 0 x̸= 0 y=λx λ ≥0x
y
x y E,
−1≤⟨x|y⟩
∥x∥∥y∥≤1
θ[0, π]
⟨x|y⟩= cos (θ)∥x∥∥y∥
cos
θ[0, π]
x y E − {0}.[ (x, y)
[
(x, y) = arccos ⟨x|y⟩
∥x∥∥y∥∈[0, π]
θ∈ {0, π},|⟨x|y⟩| =∥x∥∥y∥, x y
θ=π
2,⟨x|y⟩= 0 x, y
∥x+y∥2=∥x∥2+ 2 cos (θ)∥x∥∥y∥+∥y∥2

θ[0, π]x y.
λ, µ
\
(λx, µy) = arccos ⟨x|y⟩
∥x∥∥y∥=[
(x, y)
[0, π]
∆1=R+x1∆2=R+x2\
(∆1,∆2) = \ (x1, x2)
x1∆1x2∆2.
\
(∆1,∆2) ∆1∆2.
E u :E→
E
∀(x, y)∈E×E, ⟨u(x)|u(y)⟩=⟨x|y⟩
O(E)E.
x7→ λx Id −Id.
e∈E1,1 = ∥e∥2=∥u(e)∥2=λ2λ=±1.
E1,O(E) = {−Id, Id}.
x, y
E,
⟨x|y⟩= 0 ⇒ ⟨u(x)|u(y)⟩= 0
λ /∈ {−1,1}.
x, y E,
[
(x, y) = arccos ⟨x|y⟩
∥x∥∥y∥= arccos ⟨u(x)|u(y)⟩ ∥u(x)∥∥u(y)∥=\
(u(x), u (y))
n≥2,B= (ei)1≤i≤nE u
E E
∥u(ei)∥=∥u(ej)∥i, j 1n. λ
∥u(x)∥=λ∥x∥x∈E λ > 0, u
λ

1≤i, j ≤n, ei−ejei+ej
⟨u(ei−ej)|u(ei+ej)⟩= 0
⟨u(ei−ej)|u(ei+ej)⟩=⟨u(ei)−u(ej)|u(ei) + u(ej)⟩
=∥u(ei)∥2− ∥u(ej)∥2
∥u(ei)∥=∥u(ej)∥.
λ∥u(ei)∥.
x=
n
i=1
xiei, u (x) =
n
i=1
xiu(ei)
∥u(x)∥2=
n
i=1
x2
i∥u(ei)∥2+ 2 
1≤i<j
xixj⟨u(ei)|u(ej)⟩
=
n
i=1
x2
i∥u(ei)∥2=λ2
n
i=1
x2
i=λ2∥x∥2
⟨ei|ej⟩= 0 i̸=j⇒ ⟨u(ei)|u(ej)⟩= 0
u:E→E
∀x∈E, ∥u(x)∥=∥x∥
u
x, y
E,
⟨u(x)|u(y)⟩=1
4∥u(x) + u(y)∥2− ∥u(x)−u(y)∥2
=1
4∥u(x+y)∥2− ∥u(x−y)∥2
=1
4∥x+y∥2− ∥x−y∥2
=⟨x|y⟩
u E E
x, y E λ R,
∥u(x+λy)−u(x)−λu (y)∥2=∥u(x+λy)∥2+∥u(x)∥2+λ2∥u(y)∥2
−2 (⟨u(x+λy)|u(x)⟩+λ⟨u(x+λy)|u(y)⟩)
+ 2λ⟨u(x)|u(y)⟩
=∥x+λy∥2+∥x∥2+λ2∥y∥2
−2 (⟨x+λy |x⟩+λ⟨x+λy |y⟩)
+ 2λ⟨x|y⟩
= 2 ∥x∥2+ 2λ2∥y∥2+ 2λ⟨x|y⟩
−2∥x∥2−4λ⟨x|y⟩ − 2λ2∥y∥2
+2λ⟨x|y⟩= 0

u(x+λy) = u(x) + λu (y)u
(ei)1≤i≤nE. u
(u(ei))1≤i≤nE. x ∈E,
u(x) =
n
i=1 ⟨u(x)|u(ei)⟩u(ei) =
n
i=1 ⟨x|ei⟩u(ei)
x7→ ⟨x|ei⟩u.
−1 1.
u:E→E
e∈E
1, u :x7→ ∥x∥e u (−x) = u(x)̸=−u(x)
x̸= 0
u E E
∀(x, y)∈E×E, ∥u(x)−u(y)∥=∥x−y∥
a∈E v E u (x) = a+v(x)
x∈E u
a=u(0) a v :E→E
v(x) = u(x)−a, x ∈E. x, y E,
∥v(x)∥=∥u(x)−u(0)∥=∥x−0∥=∥x∥
∥v(x)−v(y)∥2=∥u(x)−u(y)∥2=∥x−y∥2
∥v(x)∥2−2⟨v(x)|v(y)⟩+∥v(y)∥2=∥x∥2−2⟨x|y⟩+∥y∥2
⟨v(x)|v(y)⟩=⟨x|y⟩. v
EO(E)
GL (E).
u∈ O(E). x ∈ker (u),0 = ∥u(x)∥=∥x∥x= 0.
ker (u) = {0}u u E
Id ∈ O(E)u, v O(E), x E,
∥u◦v(x)∥=∥u(v(x))∥=∥v(x)∥=∥x∥
u−1(x) = uu−1(x) =∥x∥
u◦v u−1O(E).O(E)GL (E).
EO(E)E.