Article NORMSP_2, MML version 4.99.1005

:: NORMSP_2:th 1
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Function-like quasi_total Relation of NAT,bool the carrier of b1
      st b1 is complete &
         union rng b2 = the carrier of b1 &
         (for b3 being Element of NAT holds
            (b2 . b3) ` in Family_open_set b1)
   holds ex b3 being Element of NAT st
      ex b4 being Element of REAL st
         ex b5 being Element of the carrier of b1 st
            0 < b4 & Ball(b5,b4) c= b2 . b3;

:: NORMSP_2:funcnot 1 => NORMSP_2:func 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  func distance_by_norm_of A1 -> Function-like quasi_total Relation of [:the carrier of a1,the carrier of a1:],REAL means
    for b1, b2 being Element of the carrier of a1 holds
    it .(b1,b2) = ||.b1 - b2.||;
end;

:: NORMSP_2:def 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],REAL holds
      b2 = distance_by_norm_of b1
   iff
      for b3, b4 being Element of the carrier of b1 holds
      b2 .(b3,b4) = ||.b3 - b4.||;

:: NORMSP_2:funcnot 2 => NORMSP_2:func 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  func MetricSpaceNorm A1 -> non empty Reflexive discerning symmetric triangle MetrStruct equals
    MetrStruct(#the carrier of a1,distance_by_norm_of a1#);
end;

:: NORMSP_2:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
   MetricSpaceNorm b1 = MetrStruct(#the carrier of b1,distance_by_norm_of b1#);

:: NORMSP_2:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of the carrier of MetricSpaceNorm b1
for b3 being real set holds
   ex b4 being Element of the carrier of b1 st
      b4 = b2 &
       Ball(b2,b3) = {b5 where b5 is Element of the carrier of b1: ||.b4 - b5.|| < b3};

:: NORMSP_2:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of the carrier of MetricSpaceNorm b1
for b3 being real set holds
   ex b4 being Element of the carrier of b1 st
      b4 = b2 &
       cl_Ball(b2,b3) = {b5 where b5 is Element of the carrier of b1: ||.b4 - b5.|| <= b3};

:: NORMSP_2:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of MetricSpaceNorm b1
for b4 being Element of the carrier of b1
for b5 being Element of the carrier of MetricSpaceNorm b1
      st b2 = b3 & b4 = b5
   holds    b3 is_convergent_in_metrspace_to b5
   iff
      for b6 being Element of REAL
            st 0 < b6
         holds ex b7 being Element of NAT st
            for b8 being Element of NAT
                  st b7 <= b8
               holds ||.(b2 . b8) - b4.|| < b6;

:: NORMSP_2:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of MetricSpaceNorm b1
      st b2 = b3
   holds    b3 is convergent(MetricSpaceNorm b1)
   iff
      b2 is convergent(b1);

:: NORMSP_2:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of MetricSpaceNorm b1
      st b2 = b3 & b3 is convergent(MetricSpaceNorm b1)
   holds lim b3 = lim b2;

:: NORMSP_2:funcnot 3 => NORMSP_2:func 3
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  func TopSpaceNorm A1 -> non empty TopSpace-like TopStruct equals
    TopSpaceMetr MetricSpaceNorm a1;
end;

:: NORMSP_2:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
   TopSpaceNorm b1 = TopSpaceMetr MetricSpaceNorm b1;

:: NORMSP_2:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of bool the carrier of TopSpaceNorm b1 holds
      b2 is open(TopSpaceNorm b1)
   iff
      for b3 being Element of the carrier of b1
            st b3 in b2
         holds ex b4 being Element of REAL st
            0 < b4 &
             {b5 where b5 is Element of the carrier of b1: ||.b3 - b5.|| < b4} c= b2;

:: NORMSP_2:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of the carrier of b1
for b3 being Element of REAL holds
   {b4 where b4 is Element of the carrier of b1: ||.b2 - b4.|| < b3} is open Element of bool the carrier of TopSpaceNorm b1;

:: NORMSP_2:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of the carrier of b1
for b3 being Element of REAL holds
   {b4 where b4 is Element of the carrier of b1: ||.b2 - b4.|| <= b3} is closed Element of bool the carrier of TopSpaceNorm b1;

:: NORMSP_2:th 10
theorem
for b1 being non empty TopSpace-like being_T2 TopStruct
      st b1 is locally-compact
   holds b1 is Baire;

:: NORMSP_2:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR holds
   TopSpaceNorm b1 is sequential;

:: NORMSP_2:funcreg 1
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  cluster TopSpaceNorm a1 -> non empty TopSpace-like sequential;
end;

:: NORMSP_2:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of TopSpaceNorm b1
for b4 being Element of the carrier of b1
for b5 being Element of the carrier of TopSpaceNorm b1
      st b2 = b3 & b4 = b5
   holds    b3 is_convergent_to b5
   iff
      for b6 being Element of REAL
            st 0 < b6
         holds ex b7 being Element of NAT st
            for b8 being Element of NAT
                  st b7 <= b8
               holds ||.(b2 . b8) - b4.|| < b6;

:: NORMSP_2:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of TopSpaceNorm b1
      st b2 = b3
   holds    b3 is convergent(TopSpaceNorm b1)
   iff
      b2 is convergent(b1);

:: NORMSP_2:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of TopSpaceNorm b1
      st b2 = b3 & b3 is convergent(TopSpaceNorm b1)
   holds Lim b3 = {lim b2} & lim b3 = lim b2;

:: NORMSP_2:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of bool the carrier of TopSpaceNorm b1
      st b2 = b3
   holds    b2 is closed(b1)
   iff
      b3 is closed(TopSpaceNorm b1);

:: NORMSP_2:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of bool the carrier of TopSpaceNorm b1
      st b2 = b3
   holds    b2 is open(b1)
   iff
      b3 is open(TopSpaceNorm b1);

:: NORMSP_2:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of bool the carrier of TopSpaceNorm b1
for b4 being Element of the carrier of b1
for b5 being Element of the carrier of TopSpaceNorm b1
      st b2 = b3 & b4 = b5
   holds    b2 is Neighbourhood of b4
   iff
      b3 is a_neighborhood of b5;

:: NORMSP_2:th 18
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Function-like quasi_total Relation of the carrier of TopSpaceNorm b1,the carrier of TopSpaceNorm b2
for b5 being Element of the carrier of b1
for b6 being Element of the carrier of TopSpaceNorm b1
      st b3 = b4 & b5 = b6
   holds    b3 is_continuous_in b5
   iff
      b4 is_continuous_at b6;

:: NORMSP_2:th 19
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Function-like quasi_total Relation of the carrier of TopSpaceNorm b1,the carrier of TopSpaceNorm b2
      st b3 = b4
   holds    b3 is_continuous_on the carrier of b1
   iff
      b4 is continuous(TopSpaceNorm b1, TopSpaceNorm b2);

:: NORMSP_2:funcnot 4 => NORMSP_2:func 4
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  func LinearTopSpaceNorm A1 -> non empty strict RLTopStruct means
    the carrier of it = the carrier of a1 & 0. it = 0. a1 & the addF of it = the addF of a1 & the Mult of it = the Mult of a1 & the topology of it = the topology of TopSpaceNorm a1;
end;

:: NORMSP_2:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being non empty strict RLTopStruct holds
      b2 = LinearTopSpaceNorm b1
   iff
      the carrier of b2 = the carrier of b1 & 0. b2 = 0. b1 & the addF of b2 = the addF of b1 & the Mult of b2 = the Mult of b1 & the topology of b2 = the topology of TopSpaceNorm b1;

:: NORMSP_2:funcreg 2
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  cluster LinearTopSpaceNorm a1 -> non empty TopSpace-like right_complementable Abelian add-associative right_zeroed RealLinearSpace-like strict add-continuous Mult-continuous;
end;

:: NORMSP_2:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of bool the carrier of TopSpaceNorm b1
for b3 being Element of bool the carrier of LinearTopSpaceNorm b1
      st b2 = b3
   holds    b2 is open(TopSpaceNorm b1)
   iff
      b3 is open(LinearTopSpaceNorm b1);

:: NORMSP_2:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of bool the carrier of TopSpaceNorm b1
for b3 being Element of bool the carrier of LinearTopSpaceNorm b1
      st b2 = b3
   holds    b2 is closed(TopSpaceNorm b1)
   iff
      b3 is closed(LinearTopSpaceNorm b1);

:: NORMSP_2:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of bool the carrier of LinearTopSpaceNorm b1 holds
      b2 is open(LinearTopSpaceNorm b1)
   iff
      for b3 being Element of the carrier of b1
            st b3 in b2
         holds ex b4 being Element of REAL st
            0 < b4 &
             {b5 where b5 is Element of the carrier of b1: ||.b3 - b5.|| < b4} c= b2;

:: NORMSP_2:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of the carrier of b1
for b3 being Element of REAL
for b4 being Element of bool the carrier of LinearTopSpaceNorm b1
      st b4 = {b5 where b5 is Element of the carrier of b1: ||.b2 - b5.|| < b3}
   holds b4 is open(LinearTopSpaceNorm b1);

:: NORMSP_2:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of the carrier of b1
for b3 being Element of REAL
for b4 being Element of bool the carrier of TopSpaceNorm b1
      st b4 = {b5 where b5 is Element of the carrier of b1: ||.b2 - b5.|| <= b3}
   holds b4 is closed(TopSpaceNorm b1);

:: NORMSP_2:funcreg 3
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  cluster LinearTopSpaceNorm a1 -> non empty being_T2 strict;
end;

:: NORMSP_2:funcreg 4
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  cluster LinearTopSpaceNorm a1 -> non empty sober strict;
end;

:: NORMSP_2:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of bool bool the carrier of TopSpaceNorm b1
for b3 being Element of bool bool the carrier of LinearTopSpaceNorm b1
for b4 being Element of the carrier of TopSpaceNorm b1
for b5 being Element of the carrier of LinearTopSpaceNorm b1
      st b2 = b3 & b4 = b5
   holds    b3 is Basis of b5
   iff
      b2 is Basis of b4;

:: NORMSP_2:funcreg 5
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  cluster LinearTopSpaceNorm a1 -> non empty first-countable strict;
end;

:: NORMSP_2:funcreg 6
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  cluster LinearTopSpaceNorm a1 -> non empty Frechet strict;
end;

:: NORMSP_2:funcreg 7
registration
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  cluster LinearTopSpaceNorm a1 -> non empty sequential strict;
end;

:: NORMSP_2:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of TopSpaceNorm b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of LinearTopSpaceNorm b1
for b4 being Element of the carrier of TopSpaceNorm b1
for b5 being Element of the carrier of LinearTopSpaceNorm b1
      st b2 = b3 & b4 = b5
   holds    b3 is_convergent_to b5
   iff
      b2 is_convergent_to b4;

:: NORMSP_2:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of TopSpaceNorm b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of LinearTopSpaceNorm b1
      st b2 = b3
   holds    b3 is convergent(LinearTopSpaceNorm b1)
   iff
      b2 is convergent(TopSpaceNorm b1);

:: NORMSP_2:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of TopSpaceNorm b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of LinearTopSpaceNorm b1
      st b2 = b3 & b3 is convergent(LinearTopSpaceNorm b1)
   holds Lim b2 = Lim b3 & lim b2 = lim b3;

:: NORMSP_2:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of LinearTopSpaceNorm b1
for b4 being Element of the carrier of b1
for b5 being Element of the carrier of LinearTopSpaceNorm b1
      st b2 = b3 & b4 = b5
   holds    b3 is_convergent_to b5
   iff
      for b6 being Element of REAL
            st 0 < b6
         holds ex b7 being Element of NAT st
            for b8 being Element of NAT
                  st b7 <= b8
               holds ||.(b2 . b8) - b4.|| < b6;

:: NORMSP_2:th 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of LinearTopSpaceNorm b1
      st b2 = b3
   holds    b3 is convergent(LinearTopSpaceNorm b1)
   iff
      b2 is convergent(b1);

:: NORMSP_2:th 31
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Function-like quasi_total Relation of NAT,the carrier of LinearTopSpaceNorm b1
      st b2 = b3 & b3 is convergent(LinearTopSpaceNorm b1)
   holds Lim b3 = {lim b2} & lim b3 = lim b2;

:: NORMSP_2:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of bool the carrier of LinearTopSpaceNorm b1
      st b2 = b3
   holds    b2 is closed(b1)
   iff
      b3 is closed(LinearTopSpaceNorm b1);

:: NORMSP_2:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of bool the carrier of b1
for b3 being Element of bool the carrier of LinearTopSpaceNorm b1
      st b2 = b3
   holds    b2 is open(b1)
   iff
      b3 is open(LinearTopSpaceNorm b1);

:: NORMSP_2:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of bool the carrier of TopSpaceNorm b1
for b3 being Element of bool the carrier of LinearTopSpaceNorm b1
for b4 being Element of the carrier of TopSpaceNorm b1
for b5 being Element of the carrier of LinearTopSpaceNorm b1
      st b2 = b3 & b4 = b5
   holds    b2 is a_neighborhood of b4
   iff
      b3 is a_neighborhood of b5;

:: NORMSP_2:th 35
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like quasi_total Relation of the carrier of TopSpaceNorm b1,the carrier of TopSpaceNorm b2
for b4 being Function-like quasi_total Relation of the carrier of LinearTopSpaceNorm b1,the carrier of LinearTopSpaceNorm b2
for b5 being Element of the carrier of TopSpaceNorm b1
for b6 being Element of the carrier of LinearTopSpaceNorm b1
      st b3 = b4 & b5 = b6
   holds    b3 is_continuous_at b5
   iff
      b4 is_continuous_at b6;

:: NORMSP_2:th 36
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like quasi_total Relation of the carrier of TopSpaceNorm b1,the carrier of TopSpaceNorm b2
for b4 being Function-like quasi_total Relation of the carrier of LinearTopSpaceNorm b1,the carrier of LinearTopSpaceNorm b2
      st b3 = b4
   holds    b3 is continuous(TopSpaceNorm b1, TopSpaceNorm b2)
   iff
      b4 is continuous(LinearTopSpaceNorm b1, LinearTopSpaceNorm b2);