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Theorem List for Metamath Proof Explorer - 21601-21700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhv2times 21601 Two times a vector. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( 2  .h  A )  =  ( A  +h  A ) )
 
Theoremhvnegdii 21602 Distribution of negative over subtraction. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( -u 1  .h  ( A  -h  B ) )  =  ( B  -h  A )
 
Theoremhvsubeq0i 21603 If the difference between two vectors is zero, they are equal. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  (
 ( A  -h  B )  =  0h  <->  A  =  B )
 
Theoremhvsubcan2i 21604 Vector cancellation law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  (
 ( A  +h  B )  +h  ( A  -h  B ) )  =  ( 2  .h  A )
 
Theoremhvaddcani 21605 Cancellation law for vector addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  C  e.  ~H   =>    |-  ( ( A  +h  B )  =  ( A  +h  C )  <->  B  =  C )
 
Theoremhvsubaddi 21606 Relationship between vector subtraction and addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  C  e.  ~H   =>    |-  ( ( A  -h  B )  =  C  <->  ( B  +h  C )  =  A )
 
Theoremhvnegdi 21607 Distribution of negative over subtraction. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( -u 1  .h  ( A  -h  B ) )  =  ( B  -h  A ) )
 
Theoremhvsubeq0 21608 If the difference between two vectors is zero, they are equal. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  -h  B )  =  0h  <->  A  =  B ) )
 
Theoremhvaddeq0 21609 If the sum of two vectors is zero, one is the negative of the other. (Contributed by NM, 10-Jun-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  +h  B )  =  0h  <->  A  =  ( -u 1  .h  B ) ) )
 
Theoremhvaddcan 21610 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  +h  B )  =  ( A  +h  C )  <->  B  =  C ) )
 
Theoremhvaddcan2 21611 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  +h  C )  =  ( B  +h  C )  <->  A  =  B ) )
 
Theoremhvmulcan 21612 Cancellation law for scalar multiplication. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CC  /\  A  =/=  0
 )  /\  B  e.  ~H 
 /\  C  e.  ~H )  ->  ( ( A  .h  B )  =  ( A  .h  C ) 
 <->  B  =  C ) )
 
Theoremhvmulcan2 21613 Cancellation law for scalar multiplication. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  ~H  /\  C  =/=  0h )
 )  ->  ( ( A  .h  C )  =  ( B  .h  C ) 
 <->  A  =  B ) )
 
Theoremhvsubcan 21614 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  -h  B )  =  ( A  -h  C )  <->  B  =  C ) )
 
Theoremhvsubcan2 21615 Cancellation law for vector addition. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  -h  C )  =  ( B  -h  C )  <->  A  =  B ) )
 
Theoremhvsub0 21616 Subtraction of a zero vector. (Contributed by NM, 2-Apr-2000.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( A  -h  0h )  =  A )
 
Theoremhvsubadd 21617 Relationship between vector subtraction and addition. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  -h  B )  =  C  <->  ( B  +h  C )  =  A ) )
 
Theoremhvaddsub4 21618 Hilbert vector space addition/subtraction law. (Contributed by NM, 18-May-2005.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  ~H 
 /\  B  e.  ~H )  /\  ( C  e.  ~H 
 /\  D  e.  ~H ) )  ->  ( ( A  +h  B )  =  ( C  +h  D )  <->  ( A  -h  C )  =  ( D  -h  B ) ) )
 
17.1.6  Inner product postulates for a Hilbert space
 
Axiomax-hfi 21619 Inner product maps pairs from  ~H to  CC. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
 |-  .ih  : ( ~H  X.  ~H )
 --> CC
 
Theoremhicl 21620 Closure of inner product. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  B )  e.  CC )
 
Theoremhicli 21621 Closure inference for inner product. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( A  .ih  B )  e. 
 CC
 
Axiomax-his1 21622 Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that  * `  x is the complex conjugate cjval 11553 of  x. In the literature, the inner product of  A and  B is usually written  <. A ,  B >., but our operation notation co 5792 allows us to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 3623. Physicists use  <. B  |  A >., called Dirac bra-ket notation, to represent this operation; see comments in df-bra 22391. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  B )  =  ( * `  ( B  .ih  A ) ) )
 
Axiomax-his2 21623 Distributive law for inner product. Postulate (S2) of [Beran] p. 95. (Contributed by NM, 31-Jul-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  +h  B )  .ih  C )  =  ( ( A 
 .ih  C )  +  ( B  .ih  C ) ) )
 
Axiomax-his3 21624 Associative law for inner product. Postulate (S3) of [Beran] p. 95. Warning: Mathematics textbooks usually use our version of the axiom. Physics textbooks, on the other hand, usually replace the left-hand side with  ( B  .ih  ( A  .h  C
) ) (e.g. Equation 1.21b of [Hughes] p. 44; Definition (iii) of [ReedSimon] p. 36). See the comments in df-bra 22391 for why the physics definition is swapped. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  .h  B )  .ih  C )  =  ( A  x.  ( B  .ih  C ) ) )
 
Axiomax-his4 21625 Identity law for inner product. Postulate (S4) of [Beran] p. 95. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  A  =/=  0h )  ->  0  <  ( A 
 .ih  A ) )
 
17.2  Inner product and norms
 
17.2.1  Inner product
 
Theoremhis5 21626 Associative law for inner product. Lemma 3.1(S5) of [Beran] p. 95. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  .ih  ( A  .h  C ) )  =  ( ( * `
  A )  x.  ( B  .ih  C ) ) )
 
Theoremhis52 21627 Associative law for inner product. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  .ih  (
 ( * `  A )  .h  C ) )  =  ( A  x.  ( B  .ih  C ) ) )
 
Theoremhis35 21628 Move scalar multiplication to outside of inner product. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  ~H 
 /\  D  e.  ~H ) )  ->  ( ( A  .h  C ) 
 .ih  ( B  .h  D ) )  =  ( ( A  x.  ( * `  B ) )  x.  ( C 
 .ih  D ) ) )
 
Theoremhis35i 21629 Move scalar multiplication to outside of inner product. (Contributed by NM, 1-Jul-2005.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  ~H   &    |-  D  e.  ~H   =>    |-  (
 ( A  .h  C )  .ih  ( B  .h  D ) )  =  ( ( A  x.  ( * `  B ) )  x.  ( C 
 .ih  D ) )
 
Theoremhis7 21630 Distributive law for inner product. Lemma 3.1(S7) of [Beran] p. 95. (Contributed by NM, 31-Jul-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .ih  ( B  +h  C ) )  =  ( ( A 
 .ih  B )  +  ( A  .ih  C ) ) )
 
Theoremhiassdi 21631 Distributive/associative law for inner product, useful for linearity proofs. (Contributed by NM, 10-May-2005.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  CC  /\  B  e.  ~H )  /\  ( C  e.  ~H 
 /\  D  e.  ~H ) )  ->  ( ( ( A  .h  B )  +h  C )  .ih  D )  =  ( ( A  x.  ( B 
 .ih  D ) )  +  ( C  .ih  D ) ) )
 
Theoremhis2sub 21632 Distributive law for inner product of vector subtraction. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  -h  B )  .ih  C )  =  ( ( A 
 .ih  C )  -  ( B  .ih  C ) ) )
 
Theoremhis2sub2 21633 Distributive law for inner product of vector subtraction. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .ih  ( B  -h  C ) )  =  ( ( A 
 .ih  B )  -  ( A  .ih  C ) ) )
 
Theoremhire 21634 A necessary and sufficient condition for an inner product to be real. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  .ih  B )  e.  RR  <->  ( A  .ih  B )  =  ( B 
 .ih  A ) ) )
 
Theoremhiidrcl 21635 Real closure of inner product with self. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( A  .ih  A )  e.  RR )
 
Theoremhi01 21636 Inner product with the 0 vector. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( 0h  .ih  A )  =  0 )
 
Theoremhi02 21637 Inner product with the 0 vector. (Contributed by NM, 13-Oct-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( A  .ih  0h )  =  0 )
 
Theoremhiidge0 21638 Inner product with self is not negative. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  0 
 <_  ( A  .ih  A ) )
 
Theoremhis6 21639 Zero inner product with self means vector is zero. Lemma 3.1(S6) of [Beran] p. 95. (Contributed by NM, 27-Jul-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( ( A  .ih  A )  =  0  <->  A  =  0h ) )
 
Theoremhis1i 21640 Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. (Contributed by NM, 15-May-2005.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( A  .ih  B )  =  ( * `  ( B  .ih  A ) )
 
Theoremabshicom 21641 Commuted inner products have the same absolute values. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( abs `  ( A  .ih  B ) )  =  ( abs `  ( B  .ih  A ) ) )
 
Theoremhial0 21642* A vector whose inner product is always zero is zero. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  (
 A. x  e.  ~H  ( A  .ih  x )  =  0  <->  A  =  0h ) )
 
Theoremhial02 21643* A vector whose inner product is always zero is zero. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  (
 A. x  e.  ~H  ( x  .ih  A )  =  0  <->  A  =  0h ) )
 
Theoremhisubcomi 21644 Two vector subtractions simultaneously commute in an inner product. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  C  e.  ~H   &    |-  D  e.  ~H   =>    |-  (
 ( A  -h  B )  .ih  ( C  -h  D ) )  =  ( ( B  -h  A )  .ih  ( D  -h  C ) )
 
Theoremhi2eq 21645 Lemma used to prove equality of vectors. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  .ih  ( A  -h  B ) )  =  ( B 
 .ih  ( A  -h  B ) )  <->  A  =  B ) )
 
Theoremhial2eq 21646* Two vectors whose inner product is always equal are equal. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A. x  e. 
 ~H  ( A  .ih  x )  =  ( B 
 .ih  x )  <->  A  =  B ) )
 
Theoremhial2eq2 21647* Two vectors whose inner product is always equal are equal. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A. x  e. 
 ~H  ( x  .ih  A )  =  ( x 
 .ih  B )  <->  A  =  B ) )
 
Theoremorthcom 21648 Orthogonality commutes. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  .ih  B )  =  0  <->  ( B  .ih  A )  =  0 ) )
 
Theoremnormlem0 21649 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 7-Oct-1999.) (New usage is discouraged.)
 |-  S  e.  CC   &    |-  F  e.  ~H   &    |-  G  e.  ~H   =>    |-  ( ( F  -h  ( S  .h  G ) )  .ih  ( F  -h  ( S  .h  G ) ) )  =  ( ( ( F  .ih  F )  +  ( -u ( * `  S )  x.  ( F  .ih  G ) ) )  +  ( (
 -u S  x.  ( G  .ih  F ) )  +  ( ( S  x.  ( * `  S ) )  x.  ( G  .ih  G ) ) ) )
 
Theoremnormlem1 21650 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 22-Aug-1999.) (New usage is discouraged.)
 |-  S  e.  CC   &    |-  F  e.  ~H   &    |-  G  e.  ~H   &    |-  R  e.  RR   &    |-  ( abs `  S )  =  1   =>    |-  ( ( F  -h  ( ( S  x.  R )  .h  G ) )  .ih  ( F  -h  ( ( S  x.  R )  .h  G ) ) )  =  ( ( ( F  .ih  F )  +  ( ( ( * `
  S )  x.  -u R )  x.  ( F  .ih  G ) ) )  +  ( ( ( S  x.  -u R )  x.  ( G  .ih  F ) )  +  (
 ( R ^ 2
 )  x.  ( G 
 .ih  G ) ) ) )
 
Theoremnormlem2 21651 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 27-Jul-1999.) (New usage is discouraged.)
 |-  S  e.  CC   &    |-  F  e.  ~H   &    |-  G  e.  ~H   &    |-  B  =  -u ( ( ( * `
  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )   =>    |-  B  e.  RR
 
Theoremnormlem3 21652 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)
 |-  S  e.  CC   &    |-  F  e.  ~H   &    |-  G  e.  ~H   &    |-  B  =  -u ( ( ( * `
  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )   &    |-  A  =  ( G  .ih  G )   &    |-  C  =  ( F 
 .ih  F )   &    |-  R  e.  RR   =>    |-  (
 ( ( A  x.  ( R ^ 2 ) )  +  ( B  x.  R ) )  +  C )  =  ( ( ( F 
 .ih  F )  +  (
 ( ( * `  S )  x.  -u R )  x.  ( F  .ih  G ) ) )  +  ( ( ( S  x.  -u R )  x.  ( G  .ih  F ) )  +  (
 ( R ^ 2
 )  x.  ( G 
 .ih  G ) ) ) )
 
Theoremnormlem4 21653 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
 |-  S  e.  CC   &    |-  F  e.  ~H   &    |-  G  e.  ~H   &    |-  B  =  -u ( ( ( * `
  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )   &    |-  A  =  ( G  .ih  G )   &    |-  C  =  ( F 
 .ih  F )   &    |-  R  e.  RR   &    |-  ( abs `  S )  =  1   =>    |-  ( ( F  -h  ( ( S  x.  R )  .h  G ) )  .ih  ( F  -h  ( ( S  x.  R )  .h  G ) ) )  =  ( ( ( A  x.  ( R ^ 2 ) )  +  ( B  x.  R ) )  +  C )
 
Theoremnormlem5 21654 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 10-Aug-1999.) (New usage is discouraged.)
 |-  S  e.  CC   &    |-  F  e.  ~H   &    |-  G  e.  ~H   &    |-  B  =  -u ( ( ( * `
  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )   &    |-  A  =  ( G  .ih  G )   &    |-  C  =  ( F 
 .ih  F )   &    |-  R  e.  RR   &    |-  ( abs `  S )  =  1   =>    |-  0  <_  ( (
 ( A  x.  ( R ^ 2 ) )  +  ( B  x.  R ) )  +  C )
 
Theoremnormlem6 21655 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 2-Aug-1999.) (Revised by Mario Carneiro, 4-Jun-2014.) (New usage is discouraged.)
 |-  S  e.  CC   &    |-  F  e.  ~H   &    |-  G  e.  ~H   &    |-  B  =  -u ( ( ( * `
  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )   &    |-  A  =  ( G  .ih  G )   &    |-  C  =  ( F 
 .ih  F )   &    |-  ( abs `  S )  =  1   =>    |-  ( abs `  B )  <_  ( 2  x.  ( ( sqr `  A )  x.  ( sqr `  C ) ) )
 
Theoremnormlem7 21656 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)
 |-  S  e.  CC   &    |-  F  e.  ~H   &    |-  G  e.  ~H   &    |-  ( abs `  S )  =  1   =>    |-  ( ( ( * `  S )  x.  ( F  .ih  G ) )  +  ( S  x.  ( G  .ih  F ) ) )  <_  ( 2  x.  (
 ( sqr `  ( G  .ih  G ) )  x.  ( sqr `  ( F  .ih  F ) ) ) )
 
Theoremnormlem8 21657 Lemma used to derive properties of norm. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  C  e.  ~H   &    |-  D  e.  ~H   =>    |-  (
 ( A  +h  B )  .ih  ( C  +h  D ) )  =  ( ( ( A 
 .ih  C )  +  ( B  .ih  D ) )  +  ( ( A 
 .ih  D )  +  ( B  .ih  C ) ) )
 
Theoremnormlem9 21658 Lemma used to derive properties of norm. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  C  e.  ~H   &    |-  D  e.  ~H   =>    |-  (
 ( A  -h  B )  .ih  ( C  -h  D ) )  =  ( ( ( A 
 .ih  C )  +  ( B  .ih  D ) )  -  ( ( A 
 .ih  D )  +  ( B  .ih  C ) ) )
 
Theoremnormlem7tALT 21659 Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  (
 ( S  e.  CC  /\  ( abs `  S )  =  1 )  ->  ( ( ( * `
  S )  x.  ( A  .ih  B ) )  +  ( S  x.  ( B  .ih  A ) ) )  <_  ( 2  x.  (
 ( sqr `  ( B  .ih  B ) )  x.  ( sqr `  ( A  .ih  A ) ) ) ) )
 
Theorembcseqi 21660 Equality case of Bunjakovaskij-Cauchy-Schwarz inequality. Specifically, in the equality case the two vectors are collinear. Compare bcsiHIL 21720. (Contributed by NM, 16-Jul-2001.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  (
 ( ( A  .ih  B )  x.  ( B 
 .ih  A ) )  =  ( ( A  .ih  A )  x.  ( B 
 .ih  B ) )  <->  ( ( B 
 .ih  B )  .h  A )  =  ( ( A  .ih  B )  .h  B ) )
 
Theoremnormlem9at 21661 Lemma used to derive properties of norm. Part of Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 10-May-2005.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  -h  B )  .ih  ( A  -h  B ) )  =  ( ( ( A  .ih  A )  +  ( B  .ih  B ) )  -  (
 ( A  .ih  B )  +  ( B  .ih  A ) ) ) )
 
17.2.2  Norms
 
Theoremdfhnorm2 21662 Alternate definition of the norm of a vector of Hilbert space. Definition of norm in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  normh  =  ( x  e.  ~H  |->  ( sqr `  ( x  .ih  x ) ) )
 
Theoremnormf 21663 The norm function maps from Hilbert space to reals. (Contributed by NM, 6-Sep-2007.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  normh : ~H --> RR
 
Theoremnormval 21664 The value of the norm of a vector in Hilbert space. Definition of norm in [Beran] p. 96. In the literature, the norm of  A is usually written as "||  A ||", but we use function value notation to take advantage of our existing theorems about functions. (Contributed by NM, 29-May-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  (
 normh `  A )  =  ( sqr `  ( A  .ih  A ) ) )
 
Theoremnormcl 21665 Real closure of the norm of a vector. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  (
 normh `  A )  e. 
 RR )
 
Theoremnormge0 21666 The norm of a vector is nonnegative. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  0 
 <_  ( normh `  A )
 )
 
Theoremnormgt0 21667 The norm of nonzero vector is positive. (Contributed by NM, 10-Apr-2006.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( A  =/=  0h  <->  0  <  ( normh `  A ) ) )
 
Theoremnorm0 21668 The norm of a zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
 |-  ( normh `  0h )  =  0
 
Theoremnorm-i 21669 Theorem 3.3(i) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( ( normh `  A )  =  0  <->  A  =  0h ) )
 
Theoremnormne0 21670 A norm is nonzero iff its argument is a nonzero vector. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( ( normh `  A )  =/=  0  <->  A  =/=  0h )
 )
 
Theoremnormcli 21671 Real closure of the norm of a vector. (Contributed by NM, 30-Sep-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   =>    |-  ( normh `  A )  e.  RR
 
Theoremnormsqi 21672 The square of a norm. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   =>    |-  ( ( normh `  A ) ^ 2 )  =  ( A  .ih  A )
 
Theoremnorm-i-i 21673 Theorem 3.3(i) of [Beran] p. 97. (Contributed by NM, 5-Sep-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   =>    |-  ( ( normh `  A )  =  0  <->  A  =  0h )
 
Theoremnormsq 21674 The square of a norm. (Contributed by NM, 12-May-2005.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( ( normh `  A ) ^ 2 )  =  ( A  .ih  A ) )
 
Theoremnormsub0i 21675 Two vectors are equal iff the norm of their difference is zero. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  (
 ( normh `  ( A  -h  B ) )  =  0  <->  A  =  B )
 
Theoremnormsub0 21676 Two vectors are equal iff the norm of their difference is zero. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( normh `  ( A  -h  B ) )  =  0  <->  A  =  B ) )
 
Theoremnorm-ii-i 21677 Triangle inequality for norms. Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( normh `  ( A  +h  B ) )  <_  ( ( normh `  A )  +  ( normh `  B ) )
 
Theoremnorm-ii 21678 Triangle inequality for norms. Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( normh `  ( A  +h  B ) )  <_  ( ( normh `  A )  +  ( normh `  B ) ) )
 
Theoremnorm-iii-i 21679 Theorem 3.3(iii) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)
 |-  A  e.  CC   &    |-  B  e.  ~H   =>    |-  ( normh `  ( A  .h  B ) )  =  ( ( abs `  A )  x.  ( normh `  B ) )
 
Theoremnorm-iii 21680 Theorem 3.3(iii) of [Beran] p. 97. (Contributed by NM, 25-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  CC  /\  B  e.  ~H )  ->  ( normh `  ( A  .h  B ) )  =  ( ( abs `  A )  x.  ( normh `  B ) ) )
 
Theoremnormsubi 21681 Negative doesn't change the norm of a Hilbert space vector. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( normh `  ( A  -h  B ) )  =  ( normh `  ( B  -h  A ) )
 
Theoremnormpythi 21682 Analogy to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of [Beran] p. 98. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  (
 ( A  .ih  B )  =  0  ->  ( ( normh `  ( A  +h  B ) ) ^
 2 )  =  ( ( ( normh `  A ) ^ 2 )  +  ( ( normh `  B ) ^ 2 ) ) )
 
Theoremnormsub 21683 Swapping order of subtraction doesn't change the norm of a vector. (Contributed by NM, 14-Aug-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( normh `  ( A  -h  B ) )  =  ( normh `  ( B  -h  A ) ) )
 
Theoremnormneg 21684 The norm of a vector equals the norm of its negative. (Contributed by NM, 23-May-2005.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  (
 normh `  ( -u 1  .h  A ) )  =  ( normh `  A )
 )
 
Theoremnormpyth 21685 Analogy to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of [Beran] p. 98. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  .ih  B )  =  0  ->  ( ( normh `  ( A  +h  B ) ) ^ 2 )  =  ( ( ( normh `  A ) ^ 2
 )  +  ( (
 normh `  B ) ^
 2 ) ) ) )
 
Theoremnormpyc 21686 Corollary to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of [Beran] p. 98. (Contributed by NM, 26-Oct-1999.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  .ih  B )  =  0  ->  ( normh `  A )  <_  ( normh `  ( A  +h  B ) ) ) )
 
Theoremnorm3difi 21687 Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  C  e.  ~H   =>    |-  ( normh `  ( A  -h  B ) )  <_  ( ( normh `  ( A  -h  C ) )  +  ( normh `  ( C  -h  B ) ) )
 
Theoremnorm3adifii 21688 Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 30-Sep-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  C  e.  ~H   =>    |-  ( abs `  (
 ( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) ) )  <_  ( normh `  ( A  -h  B ) )
 
Theoremnorm3lem 21689 Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  C  e.  ~H   &    |-  D  e.  RR   =>    |-  (
 ( ( normh `  ( A  -h  C ) )  <  ( D  / 
 2 )  /\  ( normh `  ( C  -h  B ) )  < 
 ( D  /  2
 ) )  ->  ( normh `  ( A  -h  B ) )  <  D )
 
Theoremnorm3dif 21690 Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 20-Apr-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( normh `  ( A  -h  B ) )  <_  ( ( normh `  ( A  -h  C ) )  +  ( normh `  ( C  -h  B ) ) ) )
 
Theoremnorm3dif2 21691 Norm of differences around common element. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( normh `  ( A  -h  B ) )  <_  ( ( normh `  ( C  -h  A ) )  +  ( normh `  ( C  -h  B ) ) ) )
 
Theoremnorm3lemt 21692 Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.)
 |-  (
 ( ( A  e.  ~H 
 /\  B  e.  ~H )  /\  ( C  e.  ~H 
 /\  D  e.  RR ) )  ->  ( ( ( normh `  ( A  -h  C ) )  < 
 ( D  /  2
 )  /\  ( normh `  ( C  -h  B ) )  <  ( D 
 /  2 ) ) 
 ->  ( normh `  ( A  -h  B ) )  <  D ) )
 
Theoremnorm3adifi 21693 Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 3-Oct-1999.) (New usage is discouraged.)
 |-  C  e.  ~H   =>    |-  ( ( A  e.  ~H 
 /\  B  e.  ~H )  ->  ( abs `  (
 ( normh `  ( A  -h  C ) )  -  ( normh `  ( B  -h  C ) ) ) )  <_  ( normh `  ( A  -h  B ) ) )
 
Theoremnormpari 21694 Parallelogram law for norms. Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  (
 ( ( normh `  ( A  -h  B ) ) ^ 2 )  +  ( ( normh `  ( A  +h  B ) ) ^ 2 ) )  =  ( ( 2  x.  ( ( normh `  A ) ^ 2
 ) )  +  (
 2  x.  ( (
 normh `  B ) ^
 2 ) ) )
 
Theoremnormpar 21695 Parallelogram law for norms. Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( ( normh `  ( A  -h  B ) ) ^ 2
 )  +  ( (
 normh `  ( A  +h  B ) ) ^
 2 ) )  =  ( ( 2  x.  ( ( normh `  A ) ^ 2 ) )  +  ( 2  x.  ( ( normh `  B ) ^ 2 ) ) ) )
 
Theoremnormpar2i 21696 Corollary of parallelogram law for norms. Part of Lemma 3.6 of [Beran] p. 100. (Contributed by NM, 5-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  C  e.  ~H   =>    |-  ( ( normh `  ( A  -h  B ) ) ^ 2 )  =  ( ( ( 2  x.  ( ( normh `  ( A  -h  C ) ) ^ 2
 ) )  +  (
 2  x.  ( (
 normh `  ( B  -h  C ) ) ^
 2 ) ) )  -  ( ( normh `  ( ( A  +h  B )  -h  (
 2  .h  C ) ) ) ^ 2
 ) )
 
Theorempolid2i 21697 Generalized polarization identity. Generalization of Exercise 4(a) of [ReedSimon] p. 63. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   &    |-  C  e.  ~H   &    |-  D  e.  ~H   =>    |-  ( A  .ih  B )  =  ( ( ( ( ( A  +h  C )  .ih  ( D  +h  B ) )  -  ( ( A  -h  C )  .ih  ( D  -h  B ) ) )  +  ( _i 
 x.  ( ( ( A  +h  ( _i 
 .h  C ) ) 
 .ih  ( D  +h  ( _i  .h  B ) ) )  -  ( ( A  -h  ( _i  .h  C ) )  .ih  ( D  -h  ( _i  .h  B ) ) ) ) ) )  / 
 4 )
 
Theorempolidi 21698 Polarization identity. Recovers inner product from norm. Exercise 4(a) of [ReedSimon] p. 63. The outermost operation is + instead of - due to our mathematicians' (rather than physicists') version of axiom ax-his3 21624. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( A  .ih  B )  =  ( ( ( ( ( normh `  ( A  +h  B ) ) ^
 2 )  -  (
 ( normh `  ( A  -h  B ) ) ^
 2 ) )  +  ( _i  x.  (
 ( ( normh `  ( A  +h  ( _i  .h  B ) ) ) ^ 2 )  -  ( ( normh `  ( A  -h  ( _i  .h  B ) ) ) ^ 2 ) ) ) )  /  4
 )
 
Theorempolid 21699 Polarization identity. Recovers inner product from norm. Exercise 4(a) of [ReedSimon] p. 63. The outermost operation is + instead of - due to our mathematicians' (rather than physicists') version of axiom ax-his3 21624. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  .ih  B )  =  ( (
 ( ( ( normh `  ( A  +h  B ) ) ^ 2
 )  -  ( (
 normh `  ( A  -h  B ) ) ^
 2 ) )  +  ( _i  x.  (
 ( ( normh `  ( A  +h  ( _i  .h  B ) ) ) ^ 2 )  -  ( ( normh `  ( A  -h  ( _i  .h  B ) ) ) ^ 2 ) ) ) )  /  4
 ) )
 
17.2.3  Relate Hilbert space to normed complex vector spaces
 
Theoremhilablo 21700 Hilbert space vector addition is an Abelian group operation. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
 |-  +h  e.  AbelOp
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