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ER
ER~x E
x
E
ϕ E ×ER
x7→ ϕ(x, y)y7→ ϕ(x, y)
∀(x, y)∈E2, ϕ(y, x) = ϕ(x, y)
∀x∈E, ϕ(x, x)>0
∀x∈E, ϕ(x, x) = 0 =⇒x= 0E
ϕ(x, y),(x|y), < x, y >, x.y
(x|y) (x|x)x2
E
E=Rm
x= (x1, . . . , xm)y= (y1, . . . , ym)
(x|y) =
m
X
i=1
xiyi

E=C0([a, b],R) [a, b]Ra<b
(f|g) = Zb
a
f(t)g(t)t
E=R[X]a<b
(P|Q) = Zb
a
P(t)Q(t)t
E F E
F E F
(·|·)
E=R2[X] (P|Q) = P(0)Q(0) + P0(0)Q0(0) + P00(0)Q00(0)
E=Mn(R) (A|B) = Pn
i=1 ai,ibi,i
E=Mn(R) (A|B) = tr(ATB)
C1([0,1],R) (f|g) = f(0)g(0) + Z1
0
f0(t)g0(t)t
E x, y ∈E
(x|y)6p(x|x)p(y|y)
x y
x= 0 x y
x6= 0
∀λ∈R,(λx +y|λx +y)>0
∀λ∈R, λ2(x|x)+2λ(x|y)+(y|y)>0
λ λ ∆60
(x|y)2−(x|x)(y|y)60
x y y =αx α ∈R
(x|y) = α(x|x)
(x|x)(y|y)=(x|x)(αx |αx) = α2(x|x)2
(x|y)2= (x|x)(y|y)
∆=0 λ0∈Rλ2
0(x|x)+2λ0(x|y)+(y|y)=0
(λ0x+y|λ0x+y)=0
λ0x+y= 0 x y
x1, . . . , xn>0x1+x2+··· +xn= 1
n
X
k=1
1
xk
>n2

[a, b]
a > 0
∀x>a, Zx
a
t
t6x−a
√ax
E
x7→ p(x|x)kxk=p(x|x)
(x|y)6kxk.kyk
(N0)k.kER+
(N1)∀x∈E, kxk= 0 ⇐⇒ x= 0
(N2)∀x∈E, ∀λ∈K,kλxk=|λ|kxk
(N3)∀x, y ∈E, kx+yk6kxk+kyk
x, y ∈Ekx+yk2= (x+y|x+y)
kx+yk2=kxk2+ 2(x|y) + kyk26kxk2+ 2|(x|y)|+kyk2
kxk2+ 2|(x|y)|+kyk26kxk2+ 2kxkkyk+kyk2= [kxk+kyk]2
kx+yk26[kxk+kyk]2
x, y ∈E
kx+yk=kxk+kyk ⇐⇒ x y
x y
∀x, y ∈E, kxk−kyk6kx−yk

x, y ∈E
(x|y) = 1
4kx+yk2− kx−yk2=1 2kx+yk2− kxk2− kyk2
E E ×E d
∀(x, y)∈E×E, d(x, y) = kx−yk
d
d E
(D0)d E ×ER+
(D1)∀x, y ∈E, d(x, y) = 0 ⇐⇒ x=y
(D2)∀x, y ∈E, d(x, y) = d(y, x)
(D3)∀x, y, z ∈E, d(x, z)6d(x, y) + d(y, z)
x, y ∈E
x y ⇐⇒ (x|y)=0
A, B E
A B ⇐⇒ ∀x∈A, ∀y∈B, (x|y) = 0
A E A
A⊥=y∈E / ∀x∈A, (x|y) = 0
A⊥E
A⊂B B⊥⊂A⊥
A⊥= (Vect A)⊥

(xi)i∈IE
∀i, j i 6=j=⇒(xi|xj)=0
∀i, j (xi|xj) = δj
i
δj
i0i6=j1i=j
T > 0E=CT(R)TR R
f, g ∈E
(f|g) = 1
TZT
0
f(t)g(t)t=1
TZT/2
−T/2
f(t)g(t)t
(E, (.|.))
ω=2π
TFE
ϕn:t7→ cos nωt n ∈Nψn:t7→ sin nωt n ∈N∗
(ϕ0|ϕ0) = 1 p∈N∗(ϕp|ϕp) = (ψp|ψp) = 1
2
p6=q(ϕp|ϕq) = 0 (ψp|ψq) = 0 p∈Nq∈N∗(ϕp|ψq)=0
p∈Nq∈N∗(ϕp|ψq) = 0 ϕpψq
p, q ∈N(p, q)6= (0,0)
C=1
TZT
0
cos pωt cos qωt t S =1
TZT
0
sin pωt sin qωt t
C+S=1p=q
0C−S= 0
F