Article LP_SPACE, MML version 4.99.1005

:: LP_SPACE:funcnot 1 => LP_SPACE:func 1
definition
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  let a2 be Element of REAL;
  func A1 rto_power A2 -> Function-like quasi_total Relation of NAT,REAL means
    for b1 being Element of NAT holds
       it . b1 = (abs (a1 . b1)) to_power a2;
end;

:: LP_SPACE:def 1
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of REAL
for b3 being Function-like quasi_total Relation of NAT,REAL holds
      b3 = b1 rto_power b2
   iff
      for b4 being Element of NAT holds
         b3 . b4 = (abs (b1 . b4)) to_power b2;

:: LP_SPACE:funcnot 2 => LP_SPACE:func 2
definition
  let a1 be Element of REAL;
  assume 1 <= a1;
  func the_set_of_RealSequences_l^ A1 -> non empty Element of bool the carrier of Linear_Space_of_RealSequences means
    for b1 being set holds
          b1 in it
       iff
          b1 in the_set_of_RealSequences & (seq_id b1) rto_power a1 is summable;
end;

:: LP_SPACE:def 2
theorem
for b1 being Element of REAL
   st 1 <= b1
for b2 being non empty Element of bool the carrier of Linear_Space_of_RealSequences holds
      b2 = the_set_of_RealSequences_l^ b1
   iff
      for b3 being set holds
            b3 in b2
         iff
            b3 in the_set_of_RealSequences & (seq_id b3) rto_power b1 is summable;

:: LP_SPACE:th 1
theorem
for b1, b2, b3 being Element of REAL
      st 0 <= b1 & b1 < b2 & 0 < b3
   holds b1 to_power b3 < b2 to_power b3;

:: LP_SPACE:th 2
theorem
for b1 being Element of REAL
   st 1 <= b1
for b2, b3 being Function-like quasi_total Relation of NAT,REAL
for b4 being Element of 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));

:: LP_SPACE:th 3
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Element of REAL
      st 1 <= b3 & b1 rto_power b3 is summable & b2 rto_power b3 is summable
   holds (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));

:: LP_SPACE:th 4
theorem
for b1 being Element of REAL
      st 1 <= b1
   holds the_set_of_RealSequences_l^ b1 is linearly-closed(Linear_Space_of_RealSequences);

:: LP_SPACE:th 5
theorem
for b1 being Element of REAL
      st 1 <= b1
   holds 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 Subspace of Linear_Space_of_RealSequences;

:: LP_SPACE:th 6
theorem
for b1 being Element of REAL
      st 1 <= b1
   holds 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;

:: LP_SPACE:th 7
theorem
for b1 being Element of REAL
      st 1 <= b1
   holds 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 non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;

:: LP_SPACE:funcnot 3 => LP_SPACE:func 3
definition
  let a1 be Element of REAL;
  func l_norm^ A1 -> Function-like quasi_total Relation of the_set_of_RealSequences_l^ a1,REAL means
    for b1 being set
          st b1 in the_set_of_RealSequences_l^ a1
       holds it . b1 = (Sum ((seq_id b1) rto_power a1)) to_power (1 / a1);
end;

:: LP_SPACE:def 3
theorem
for b1 being Element of REAL
for b2 being Function-like quasi_total Relation of the_set_of_RealSequences_l^ b1,REAL holds
      b2 = l_norm^ b1
   iff
      for b3 being set
            st b3 in the_set_of_RealSequences_l^ b1
         holds b2 . b3 = (Sum ((seq_id b3) rto_power b1)) to_power (1 / b1);

:: LP_SPACE:th 8
theorem
for b1 being Element of REAL
      st 1 <= b1
   holds 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#) is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;

:: LP_SPACE:th 9
theorem
for b1 being Element of REAL
      st 1 <= b1
   holds 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#) is Subspace of Linear_Space_of_RealSequences;

:: LP_SPACE:th 10
theorem
for b1 being Element of REAL
   st 1 <= b1
for b2 being non empty NORMSTR
      st 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#)
   holds the carrier of b2 = the_set_of_RealSequences_l^ b1 &
    (for b3 being set holds
          b3 is Element of the carrier of b2
       iff
          b3 is Function-like quasi_total Relation of NAT,REAL & (seq_id b3) rto_power b1 is summable) &
    0. b2 = Zeroseq &
    (for b3 being Element of the carrier of b2 holds
       b3 = seq_id b3) &
    (for b3, b4 being Element of the carrier of b2 holds
    b3 + b4 = (seq_id b3) + seq_id b4) &
    (for b3 being Element of REAL
    for b4 being Element of the carrier of b2 holds
       b3 * b4 = b3 (#) seq_id b4) &
    (for b3 being Element of the carrier of b2 holds
       - b3 = - seq_id b3 & seq_id - b3 = - seq_id b3) &
    (for b3, b4 being Element of the carrier of b2 holds
    b3 - b4 = (seq_id b3) - seq_id b4) &
    (for b3 being Element of the carrier of b2 holds
       (seq_id b3) rto_power b1 is summable) &
    (for b3 being Element of the carrier of b2 holds
       ||.b3.|| = (Sum ((seq_id b3) rto_power b1)) to_power (1 / b1));

:: LP_SPACE:th 11
theorem
for b1 being Element of REAL
   st 1 <= b1
for b2 being Function-like quasi_total Relation of NAT,REAL
      st for b3 being Element of NAT holds
           b2 . b3 = 0
   holds b2 rto_power b1 is summable &
    (Sum (b2 rto_power b1)) to_power (1 / b1) = 0;

:: LP_SPACE:th 12
theorem
for b1 being Element of REAL
   st 1 <= b1
for b2 being Function-like quasi_total Relation of NAT,REAL
   st b2 rto_power b1 is summable &
      (Sum (b2 rto_power b1)) to_power (1 / b1) = 0
for b3 being natural set holds
   b2 . b3 = 0;

:: LP_SPACE:th 13
theorem
for b1 being Element of REAL
   st 1 <= b1
for b2 being non empty NORMSTR
   st 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#)
for b3, b4 being Element of the carrier of b2
for b5 being Element of REAL holds
   (||.b3.|| = 0 implies b3 = 0. b2) &
    (b3 = 0. b2 implies ||.b3.|| = 0) &
    0 <= ||.b3.|| &
    ||.b3 + b4.|| <= ||.b3.|| + ||.b4.|| &
    ||.b5 * b3.|| = (abs b5) * ||.b3.||;

:: LP_SPACE:th 14
theorem
for b1 being Element of REAL
   st 1 <= b1
for b2 being non empty NORMSTR
      st 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#)
   holds b2 is RealNormSpace-like;

:: LP_SPACE:th 15
theorem
for b1 being Element of REAL
   st 1 <= b1
for b2 being non empty NORMSTR
      st 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#)
   holds b2 is non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;

:: LP_SPACE:th 16
theorem
for b1 being Element of REAL
   st 1 <= b1
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
   st 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#)
for b3 being Function-like quasi_total Relation of NAT,the carrier of b2
      st b3 is CCauchy(b2)
   holds b3 is convergent(b2);

:: LP_SPACE:funcnot 4 => LP_SPACE:func 4
definition
  let a1 be Element of REAL;
  assume 1 <= a1;
  func l_Space^ A1 -> non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like complete NORMSTR equals
    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#);
end;

:: LP_SPACE:def 4
theorem
for b1 being Element of REAL
      st 1 <= b1
   holds l_Space^ b1 = 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#);