:: HOLDER_1 semantic presentation
theorem Th1: :: HOLDER_1:1
theorem Th2: :: HOLDER_1:2
theorem Th3: :: HOLDER_1:3
theorem Th4: :: HOLDER_1:4
for b
1, b
2, b
3, b
4 being
Real holds
( 1
< b
1 &
(1 / b1) + (1 / b2) = 1 & 0
< b
3 & 0
< b
4 implies ( b
3 * b
4 <= ((b3 #R b1) / b1) + ((b4 #R b2) / b2) & ( b
3 * b
4 = ((b3 #R b1) / b1) + ((b4 #R b2) / b2) implies b
3 #R b
1 = b
4 #R b
2 ) & ( b
3 #R b
1 = b
4 #R b
2 implies b
3 * b
4 = ((b3 #R b1) / b1) + ((b4 #R b2) / b2) ) ) )
theorem Th5: :: HOLDER_1:5
Lemma4:
for b1 being Real_Sequence holds
( ( for b2 being Nat holds 0 <= b1 . b2 ) implies for b2 being Nat holds b1 . b2 <= (Partial_Sums b1) . b2 )
Lemma5:
for b1 being Real_Sequence holds
( ( for b2 being Nat holds 0 <= b1 . b2 ) implies for b2 being Nat holds 0 <= (Partial_Sums b1) . b2 )
Lemma6:
for b1 being Real_Sequence holds
( ( for b2 being Nat holds 0 <= b1 . b2 ) implies for b2 being Nat holds
( (Partial_Sums b1) . b2 = 0 implies for b3 being Nat holds
( b3 <= b2 implies b1 . b3 = 0 ) ) )
Lemma7:
for b1 being Real_Sequence
for b2 being Nat holds
( ( for b3 being Nat holds
( b3 <= b2 implies b1 . b3 = 0 ) ) implies (Partial_Sums b1) . b2 = 0 )
theorem Th6: :: HOLDER_1:6
theorem Th7: :: HOLDER_1:7
theorem Th8: :: HOLDER_1:8
theorem Th9: :: HOLDER_1:9
theorem Th10: :: HOLDER_1:10
theorem Th11: :: HOLDER_1:11
theorem Th12: :: HOLDER_1:12
theorem Th13: :: HOLDER_1:13