:: LP_SPACE semantic presentation
:: deftheorem Def1 defines rto_power LP_SPACE:def 1 :
:: deftheorem Def2 defines the_set_of_RealSequences_l^ LP_SPACE:def 2 :
Lemma3:
for b1, b2 being Real holds
( b1 >= 0 & b2 > 0 implies b1 to_power b2 >= 0 )
Lemma4:
for b1, b2, b3 being Real holds
( b1 > 0 & b2 >= 0 & b3 >= 0 implies (b2 * b3) to_power b1 = (b2 to_power b1) * (b3 to_power b1) )
theorem Th1: :: LP_SPACE:1
Lemma6:
for b1 being Real holds
( 1 = b1 implies for b2, b3 being Real_Sequence
for b4 being Nat holds ((Partial_Sums ((b2 + b3) rto_power b1)) . b4) to_power (1 / b1) <= (((Partial_Sums (b2 rto_power b1)) . b4) to_power (1 / b1)) + (((Partial_Sums (b3 rto_power b1)) . b4) to_power (1 / b1)) )
theorem Th2: :: LP_SPACE:2
Lemma8:
for b1, b2 being Real_Sequence
for b3 being Real holds
( 1 = b3 & b1 rto_power b3 is summable & b2 rto_power b3 is summable implies ( (b1 + b2) rto_power b3 is summable & (Sum ((b1 + b2) rto_power b3)) to_power (1 / b3) <= ((Sum (b1 rto_power b3)) to_power (1 / b3)) + ((Sum (b2 rto_power b3)) to_power (1 / b3)) ) )
theorem Th3: :: LP_SPACE:3
theorem Th4: :: LP_SPACE:4
theorem Th5: :: LP_SPACE:5
theorem Th6: :: LP_SPACE:6
for b
1 being
Real holds
( 1
<= b
1 implies (
RLSStruct(#
(the_set_of_RealSequences_l^ b1),
(Zero_ (the_set_of_RealSequences_l^ b1),Linear_Space_of_RealSequences ),
(Add_ (the_set_of_RealSequences_l^ b1),Linear_Space_of_RealSequences ),
(Mult_ (the_set_of_RealSequences_l^ b1),Linear_Space_of_RealSequences ) #) is
Abelian &
RLSStruct(#
(the_set_of_RealSequences_l^ b1),
(Zero_ (the_set_of_RealSequences_l^ b1),Linear_Space_of_RealSequences ),
(Add_ (the_set_of_RealSequences_l^ b1),Linear_Space_of_RealSequences ),
(Mult_ (the_set_of_RealSequences_l^ b1),Linear_Space_of_RealSequences ) #) is
add-associative &
RLSStruct(#
(the_set_of_RealSequences_l^ b1),
(Zero_ (the_set_of_RealSequences_l^ b1),Linear_Space_of_RealSequences ),
(Add_ (the_set_of_RealSequences_l^ b1),Linear_Space_of_RealSequences ),
(Mult_ (the_set_of_RealSequences_l^ b1),Linear_Space_of_RealSequences ) #) is
right_zeroed &
RLSStruct(#
(the_set_of_RealSequences_l^ b1),
(Zero_ (the_set_of_RealSequences_l^ b1),Linear_Space_of_RealSequences ),
(Add_ (the_set_of_RealSequences_l^ b1),Linear_Space_of_RealSequences ),
(Mult_ (the_set_of_RealSequences_l^ b1),Linear_Space_of_RealSequences ) #) is
right_complementable &
RLSStruct(#
(the_set_of_RealSequences_l^ b1),
(Zero_ (the_set_of_RealSequences_l^ b1),Linear_Space_of_RealSequences ),
(Add_ (the_set_of_RealSequences_l^ b1),Linear_Space_of_RealSequences ),
(Mult_ (the_set_of_RealSequences_l^ b1),Linear_Space_of_RealSequences ) #) is
RealLinearSpace-like ) )
by Th5;
theorem Th7: :: LP_SPACE:7
Lemma14:
for b1 being Real holds
ex b2 being Function of the_set_of_RealSequences_l^ b1, REAL st
for b3 being set holds
( b3 in the_set_of_RealSequences_l^ b1 implies b2 . b3 = (Sum ((seq_id b3) rto_power b1)) to_power (1 / b1) )
:: deftheorem Def3 defines l_norm^ LP_SPACE:def 3 :
theorem Th8: :: LP_SPACE:8
theorem Th9: :: LP_SPACE:9
theorem Th10: :: LP_SPACE:10
theorem Th11: :: LP_SPACE:11
theorem Th12: :: LP_SPACE:12
Lemma21:
for b1 being Real holds
( 1 <= b1 implies for b2 being non empty NORMSTR holds
( b2 = NORMSTR(# (the_set_of_RealSequences_l^ b1),(Zero_ (the_set_of_RealSequences_l^ b1),Linear_Space_of_RealSequences ),(Add_ (the_set_of_RealSequences_l^ b1),Linear_Space_of_RealSequences ),(Mult_ (the_set_of_RealSequences_l^ b1),Linear_Space_of_RealSequences ),(l_norm^ b1) #) implies for b3 being Point of b2
for b4 being Real holds (seq_id (b4 * b3)) rto_power b1 = ((abs b4) to_power b1) (#) ((seq_id b3) rto_power b1) ) )
Lemma22:
for b1 being Real holds
( 1 <= b1 implies for b2 being non empty NORMSTR holds
( b2 = NORMSTR(# (the_set_of_RealSequences_l^ b1),(Zero_ (the_set_of_RealSequences_l^ b1),Linear_Space_of_RealSequences ),(Add_ (the_set_of_RealSequences_l^ b1),Linear_Space_of_RealSequences ),(Mult_ (the_set_of_RealSequences_l^ b1),Linear_Space_of_RealSequences ),(l_norm^ b1) #) implies for b3 being Point of b2
for b4 being Real holds Sum ((seq_id (b4 * b3)) rto_power b1) = ((abs b4) to_power b1) * (Sum ((seq_id b3) rto_power b1)) ) )
Lemma23:
for b1 being Real holds
( 1 <= b1 implies for b2 being non empty NORMSTR holds
( b2 = NORMSTR(# (the_set_of_RealSequences_l^ b1),(Zero_ (the_set_of_RealSequences_l^ b1),Linear_Space_of_RealSequences ),(Add_ (the_set_of_RealSequences_l^ b1),Linear_Space_of_RealSequences ),(Mult_ (the_set_of_RealSequences_l^ b1),Linear_Space_of_RealSequences ),(l_norm^ b1) #) implies for b3 being Point of b2
for b4 being Real holds (Sum ((seq_id (b4 * b3)) rto_power b1)) to_power (1 / b1) = (abs b4) * ((Sum ((seq_id b3) rto_power b1)) to_power (1 / b1)) ) )
theorem Th13: :: LP_SPACE:13
theorem Th14: :: LP_SPACE:14
theorem Th15: :: LP_SPACE:15
Lemma27:
for b1 being Real
for b2 being Real_Sequence holds
( b2 is convergent implies for b3 being Real_Sequence holds
( ( for b4 being Nat holds b3 . b4 = abs ((b2 . b4) - b1) ) implies ( b3 is convergent & lim b3 = abs ((lim b2) - b1) ) ) )
by RSSPACE3:1;
Lemma28:
for b1 being Real holds
( 0 < b1 implies for b2 being Real
for b3 being Real_Sequence holds
( b3 is convergent implies for b4 being Real_Sequence holds
( ( for b5 being Nat holds b4 . b5 = (abs ((b3 . b5) - b2)) to_power b1 ) implies ( b4 is convergent & lim b4 = (abs ((lim b3) - b2)) to_power b1 ) ) ) )
Lemma29:
for b1 being Real holds
( 0 < b1 implies for b2 being Real
for b3, b4 being Real_Sequence holds
( b3 is convergent & b4 is convergent implies for b5 being Real_Sequence holds
( ( for b6 being Nat holds b5 . b6 = ((abs ((b3 . b6) - b2)) to_power b1) + (b4 . b6) ) implies ( b5 is convergent & lim b5 = ((abs ((lim b3) - b2)) to_power b1) + (lim b4) ) ) ) )
Lemma30:
for b1 being Real
for b2 being Real_Sequence holds
( b2 is convergent & ex b3 being Nat st
for b4 being Nat holds
( b3 <= b4 implies b2 . b4 <= b1 ) implies lim b2 <= b1 )
by RSSPACE2:6;
Lemma31:
for b1, b2 being Real_Sequence holds b1 = (b1 + b2) - b2
Lemma32:
for b1 being Real holds
( b1 >= 1 implies for b2, b3 being Real_Sequence holds
( b2 rto_power b1 is summable & b3 rto_power b1 is summable implies (b2 + b3) rto_power b1 is summable ) )
theorem Th16: :: LP_SPACE:16
definition
let c
1 be
Real;
assume E34:
1
<= c
1
;
func l_Space^ c
1 -> RealBanachSpace equals :: LP_SPACE:def 4
NORMSTR(#
(the_set_of_RealSequences_l^ a1),
(Zero_ (the_set_of_RealSequences_l^ a1),Linear_Space_of_RealSequences ),
(Add_ (the_set_of_RealSequences_l^ a1),Linear_Space_of_RealSequences ),
(Mult_ (the_set_of_RealSequences_l^ a1),Linear_Space_of_RealSequences ),
(l_norm^ a1) #);
coherence
NORMSTR(# (the_set_of_RealSequences_l^ c1),(Zero_ (the_set_of_RealSequences_l^ c1),Linear_Space_of_RealSequences ),(Add_ (the_set_of_RealSequences_l^ c1),Linear_Space_of_RealSequences ),(Mult_ (the_set_of_RealSequences_l^ c1),Linear_Space_of_RealSequences ),(l_norm^ c1) #) is RealBanachSpace
end;
:: deftheorem Def4 defines l_Space^ LP_SPACE:def 4 :