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E
n
(E
n
, g)
g : E
n
× E
n
3 {x, y} 7→ g(x, y) = (x, y) ∈ R,
E
n
g(x, y) = g(y, x) x, y ∈ E
n
E
n
3 x 7→ g(x, x) ∈ R
g(x, x) > 0 x ∈ E
n
g(x, x) =
0 ⇐⇒ x = 0
(E
n
, g) E
m
⊂ E
n
g
0
g E
m
g
0
(x, y) = g(x, y) x, y ∈ E
m
(E
m
, g
0
)
g
0
(x, x) = g(x, x) > 0 g
0
(x, x) = 0 ⇐⇒
x = 0
(E
n
, g) (E
0
n
, g
0
)
ϕ : E
n
→ E
0
n
g
0
(ϕ(x), ϕ(y)) = g(x, y) ∀ x, y ∈ E
n
.
ϕ :
E
n
→ E
0
n
(E
n
, g) (E
0
n
, g
0
)
(E
n
, g)
(E
0
n
, g
0
)
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bgm efghifjbhikg En w jhhgaafgkjbbgm h mkvoangkcd kmvigfbcd efghifjbp
hikgd (En, g)
ì060,*,58 -3:5
g : En × En 3 {x, y} 7→ g(x, y) = (x, y) ∈ R,
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bgu 060 dmifasmhvgu -3:- *B160D-B5 .3-/735,/7B5 En < ³75 -3:5 .-
-.3*D*6*,0 2D-B6*7B-38*7 DB2: 2/6-B08:~ -,5 /0::*730),5 7- */7>
)
g(x, y) = g(y, x) D68 6+AC x, y ∈ En G 1B5D3570 ,58 -3:5
En 3 x 7→ g(x, x) ∈ R
.-6-07*6>,- -.3*D*6*,5 7- */7> g(x, x) > 0 .30 6+-: x ∈ En 0 g(x, x) =
0 ⇐⇒ x = 0 <
XYZ[¡Y\]Y^ }hi (E , g) mkvoangkg efghifjbhikgw j E ⊂ E
egnefghifjbhikg a g0 gxfjbasmbam `gfdc g bj egnefghifjbhikg Em w
n m n
ig mhi g0(x, y) = g(x, y) not o{|cr x, y ∈ Em ¢gxnj (Em, g0) mkvoangkg
efghifjbhikg
£¤Y[¥ -)*B0D,-~ g0(x, x) = g(x, x) > 0 0 g0(x, x) = 0 ⇐⇒
x = 0<
VWXYZY[Y\]Y^ }hi (E , g) a (E0 , g0) nkj mkvoangkj kmvigfbcr efgp
hifjbhikj Ølgdgf`ald kmvigfbcr efghifjbhik ϕ : En → E0n bjlckjp
n n
miht algdgf`aldgd mkvoangkcr kmvigfbcr efghifjbhikw mhoa
g 0 (ϕ(x), ϕ(y)) = g(x, y) ∀ x, y ∈ En .
Æhoa h}qmhik}mi algdgf`ald mkvoangkcr kmvigfbcr efghifjbhik ϕ :
v n v
En → E0n w ig mk oa gkc km igfbcm efghifjbhikj (En , g) a (E0n , g 0 ) bjlcp
kj{iht algdgf`bcda
XYZ[¡Y\]Y^ í{|cm nkj mkvoangkcr kmvigfbcr efghifjbhikj (E , g)
a (E0n, g0) gnbgu fjldmfbghia algdgf`bc
n
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