244

E
x, y E
|⟨x|y⟩| ≤ ∥x∥∥y∥
x y
Rn
∀(x, y)∈Rn×Rn,
n
k=1
xkyk
2
≤n
k=1
x2
k n
k=1
y2
k
n≥1x1,··· , xn.
n
k=1
xk2 ≤n
n
k=1
x2
k
a b,
φ: (x, y)7→ a
n
i=1
xiyi+b
1≤i̸=j≤n
xiyjRn, n ≥2.
n
k=1
xk·12 ≤n
k=1
12 n
k=1
x2
k=n
n
k=1
x2
k
xk

φ x ∈Rn,
q(x) = φ(x, x) = a
n
i=1
x2
i+ 2b
1≤i<j≤n
xixj
=a
n
i=1
x2
i+b
n
i=1
xi2 −
n
i=1
x2
i
= (a−b)
n
i=1
x2
i+bn
i=1
xi2
φ a =q(e1)>0, a −b=q(e1−e2)>0
qn
i=1
ei=n(a+ (n−1) b)>0.
a > 0, a −b > 0a+ (n−1) b > 0, x ∈Rn\ {0}
q(x) = (a−b)
n
i=1
x2
i+bn
i=1
xi2
>(a−b)1
nn
i=1
xi2 +bn
i=1
xi2
=1
n(a+ (n−1) b)n
i=1
xi2 ≥0
q(x)>0. x ∈Rn\ {0}λ̸= 0,
q(x) = qλ
n
i=1
ei=nλ2(a+ (n−1) b)>0
φ
n≥1,
n
k=1
k√k≤n(n+ 1)
2√3√2n+ 1
n
k=1
k√k≤
n
k=1
k2
n
k=1
k
n
k=1
k=n(n+ 1)
2
n
k=1
k2=n(n+ 1) (2n+ 1)
6,
n
k=1
k√k≤n2(n+ 1)2(2n+ 1)
12 =n(n+ 1)
2√3√2n+ 1

n≥1x1,··· , xn
n
k=1
xk n
k=1
1
xk≥n2
n
k=1
1
k2≥6n (n+ 1) (2n+ 1)
n=
n
k=1
√xk
1
√xk≤
n
k=1
xk
n
k=1
1
xk
λ√xk=λ1
√xk
k1n, xk=λ k 1n, λ
xk=k2k1n,
n
k=1
k2 n
k=1
1
k2≥n2
n
k=1
k2=n(n+ 1) (2n+ 1)
6,
n
k=1
1
k2≥6n (n+ 1) (2n+ 1).
a, b λ,
(2λ−1) a2−2λab =λ(a−b)2−λb2+ (λ−1) a2
q E =Rn
q(x) =
n
k=1
(2k−1) x2
k−2
n−1
k=1
kxkxk+1
q
q.

(x, y) = (x1,··· , xn, y1,··· , yn)H=R2nQ
H
Q(x, y) =
n
k=1 y2
k−2xkyk.
Q
Q.
n≥1x= (x1,··· , xn)Rn, y = (y1,··· , yn)
yk=1
k
k
j=1
xj.
x1=y1
∀k∈ {2,··· , n}, xk=kyk−(k−1) yk−1
Q(x, y) = −q(y).
n
k=1
y2
k≤
n
k=1
2xkyk.
n
k=1
y2
k≤4
n
k=1
x2
k.
(xn)n≥1x2
n
(yn)n≥1yn=1
n
n
j=1
xj
n≥1,y2
n
+∞
n=1
y2
n≤4
+∞
n=1
x2
n.
(2λ−1) a2−2λab =λa2−2ab+ (λ−1) a2
=λ(a−b)2−b2+ (λ−1) a2
=λ(a−b)2−λb2+ (λ−1) a2

q(x) = (2n−1) x2
n+
n−1
k=1 (2k−1) x2
k−2kxkxk+1
= (2n−1) x2
n+
n−1
k=1
k(xk−xk+1)2−
n−1
k=1
kx2
k+1 +
n−1
k=1
(k−1) x2
k
= (2n−1) x2
n+
n−1
k=1
k(xk−xk+1)2+
n−2
k=1
kx2
k+1 −
n−1
k=1
kx2
k+1
= (2n−1) x2
n+
n−1
k=1
k(xk−xk+1)2−(n−1) x2
n
=
n−1
k=1
k(xk−xk+1)2+nx2
n.
q(x) =
n
k=1
kℓ2
k(x)
ℓk
ℓk(x) = xk−xk+1 (1 ≤k≤n−1)
ℓn(x) = xn
rg (q) = n, ker (q) = {0}sgn (q) = (n, 0) .
Q(x, y) =
n
k=1
(yk−xk)2−
n
k=1
x2
k
Q(x, y) =
n
k=1
L2
k(x, y)−
2n
k=n+1
L2
k(x, y)
Lk
Lk(x, y) = yk−xk(1 ≤k≤n)
Lk(x, y) = xk(n+ 1 ≤k≤2n)
rg (Q) = 2n, ker (Q) = {0}sgn (Q) = (n, n).
x1=y1k≥2, kyk=
k
j=1
xj,
xk=kyk−(k−1) yk−1.