x, y, z ∈X d X d(x, y)6d(x, z) + d(z, y)
d(x, y)−d(x, z)6d(z, y) = d(y, z)d(x, z)6d(x, y) + d(y, z)−d(y, z)6
d(x, y)−d(x, z)
u∈Rni∈J1, nKuii u
d1Rn
x, y ∈Rnx=y d1(x, y) = 0 d(x, y)=0
|xi−yi|= 0 iinJ1, nKx=y
x, y ∈Rn|xi−yi|=|yi−xi|i∈J1, nKd1(x, y) = d1(y, x)
x, y, z ∈Rn|xi−zi|6|xi−yi|+|yi−zi|
i∈J1, nKd1(x, z)6d1(x, y) + d1(y, z)
d2Rn(· | ·)
Rnu, v ∈Rn
(u|v) =
n
P
i=1
uivi
u∈Rn(u|u)>0kuk2=p(u|u)u, v ∈Rn
|(u|v)|6kuk2· kvk2
ku+vk26kuk2+kvk2
d2(x, y) = kx−yk2x, y ∈Rn
(· | ·)d2(x, z)6d2(x, y) +
d2(y, z)x, y, z ∈Rnu=x−y v =y−z
Rn
u, v ∈Rnv v
λ∈R
06(u+λv |u+λv)=(u|u)+2λ(u|v) + λ2(v|v)
(u|u)+2X(u|v) + X2(v|v)R[X]
4(u|v)2−4(u|u)(v|v)60
u, v ∈Rn
ku+vk2
2= (u+v|u+v)=(u|u)+2(u|v)+(v|v)6kuk2
2+2kuk2·kvk2+kvk2
2= (kuk2+kuk2)2
d∞Rn
x, y ∈Rnx=y d∞(x, y) = 0 x6=y
j∈J1, nKxj6=yjd∞(x, y)>|xj−yj|>0
x, y ∈Rn|xi−yi|=|yi−xi|i∈J1, nKd∞(x, y) = d∞(y, x)
x, y, z ∈Rnj∈J1, nK|xj−zj|= max{|xi−zi| | i∈J1, nK}
d∞
d∞(x, z) = |xj−zj|6|xj−yj|+|yj−zj|6d∞(x, y) + d∞(y, z)