:: VECTMETR semantic presentation
:: deftheorem Def1 defines convex VECTMETR:def 1 :
:: deftheorem Def2 defines internal VECTMETR:def 2 :
theorem Th1: :: VECTMETR:1
:: deftheorem Def3 defines isometric VECTMETR:def 3 :
:: deftheorem Def4 defines ISOM VECTMETR:def 4 :
theorem Th2: :: VECTMETR:2
theorem Th3: :: VECTMETR:3
theorem Th4: :: VECTMETR:4
theorem Th5: :: VECTMETR:5
registration
let X be non
empty set ;
let F be
Function of
[:X,X:],
REAL ;
let O be
Element of
X;
let B be
BinOp of
X;
let G be
Function of
[:REAL ,X:],
X;
cluster RLSMetrStruct(#
a1,
a2,
a3,
a4,
a5 #)
-> non
empty ;
coherence
not RLSMetrStruct(# X,F,O,B,G #) is empty
by STRUCT_0:def 1;
end;
:: deftheorem Def5 defines homogeneous VECTMETR:def 5 :
:: deftheorem Def6 defines translatible VECTMETR:def 6 :
:: deftheorem Def7 defines Norm VECTMETR:def 7 :
theorem Th6: :: VECTMETR:6
theorem Th7: :: VECTMETR:7
theorem Th8: :: VECTMETR:8
definition
let n be
Element of
NAT ;
func RLMSpace c1 -> strict RealLinearMetrSpace means :
Def8:
:: VECTMETR:def 8
( the
carrier of
it = REAL n & the
distance of
it = Pitag_dist n & the
Zero of
it = 0* n & ( for
x,
y being
Element of
REAL n holds the
add of
it . x,
y = x + y ) & ( for
x being
Element of
REAL n for
r being
Element of
REAL holds the
Mult of
it . r,
x = r * x ) );
existence
ex b1 being strict RealLinearMetrSpace st
( the carrier of b1 = REAL n & the distance of b1 = Pitag_dist n & the Zero of b1 = 0* n & ( for x, y being Element of REAL n holds the add of b1 . x,y = x + y ) & ( for x being Element of REAL n
for r being Element of REAL holds the Mult of b1 . r,x = r * x ) )
uniqueness
for b1, b2 being strict RealLinearMetrSpace st the carrier of b1 = REAL n & the distance of b1 = Pitag_dist n & the Zero of b1 = 0* n & ( for x, y being Element of REAL n holds the add of b1 . x,y = x + y ) & ( for x being Element of REAL n
for r being Element of REAL holds the Mult of b1 . r,x = r * x ) & the carrier of b2 = REAL n & the distance of b2 = Pitag_dist n & the Zero of b2 = 0* n & ( for x, y being Element of REAL n holds the add of b2 . x,y = x + y ) & ( for x being Element of REAL n
for r being Element of REAL holds the Mult of b2 . r,x = r * x ) holds
b1 = b2
end;
:: deftheorem Def8 defines RLMSpace VECTMETR:def 8 :
theorem Th9: :: VECTMETR:9
:: deftheorem Def9 defines IsomGroup VECTMETR:def 9 :
theorem Th10: :: VECTMETR:10
theorem Th11: :: VECTMETR:11
:: deftheorem Def10 defines SubIsomGroupRel VECTMETR:def 10 :