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Theorem List for Metamath Proof Explorer - 21501-21600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhis2sub 21501 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 21502 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 21503 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 21504 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 21505 Inner product with the 0 vector. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( 0h  .ih  A )  =  0 )
 
Theoremhi02 21506 Inner product with the 0 vector. (Contributed by NM, 13-Oct-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  ( A  .ih  0h )  =  0 )
 
Theoremhiidge0 21507 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 21508 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 21509 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 21510 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 21511* 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 21512* 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 21513 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 21514 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 21515* 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 21516* 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 21517 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 21518 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 21519 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 21520 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 21521 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 21522 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 21523 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 21524 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 21525 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 21526 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 21527 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 21528 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 21529 Equality case of Bunjakovaskij-Cauchy-Schwarz inequality. Specifically, in the equality case the two vectors are collinear. Compare bcsiHIL 21589. (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 21530 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 ) ) ) )
 
15.9.8  Norms
 
Theoremdfhnorm2 21531 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 21532 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 21533 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 21534 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 21535 The norm of a vector is nonnegative. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
 |-  ( A  e.  ~H  ->  0 
 <_  ( normh `  A )
 )
 
Theoremnormgt0 21536 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 21537 The norm of a zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
 |-  ( normh `  0h )  =  0
 
Theoremnorm-i 21538 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 21539 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 21540 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 21541 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 21542 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 21543 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 21544 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 21545 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 21546 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 21547 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 21548 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 21549 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 21550 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 21551 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 21552 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 21553 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 21554 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 21555 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 21556 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 21557 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 21558 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 21559 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 21560 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 21561 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 21562 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 21563 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 21564 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 21565 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 21566 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 21567 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 21493. (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 21568 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 21493. (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
 ) )
 
15.9.9  Relate Hilbert space to normed complex vector spaces
 
Theoremhilablo 21569 Hilbert space vector addition is an Abelian group operation. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
 |-  +h  e.  AbelOp
 
Theoremhilid 21570 The group identity element of Hilbert space vector addition is the zero vector. (Contributed by NM, 16-Apr-2007.) (New usage is discouraged.)
 |-  (GId ` 
 +h  )  =  0h
 
Theoremhilvc 21571 Hilbert space is a complex vector space. Vector addition is  +h, and scalar product is  .h. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
 |-  <.  +h  ,  .h  >.  e.  CVec OLD
 
Theoremhilnormi 21572 Hilbert space norm in terms of vector space norm. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  ~H  =  ( BaseSet `  U )   &    |-  .ih  =  ( .i OLD `  U )   &    |-  U  e.  NrmCVec   =>    |- 
 normh  =  ( normCV `  U )
 
Theoremhilhhi 21573 Deduce the structure of Hilbert space from its components. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  ~H  =  ( BaseSet `  U )   &    |-  +h  =  ( +v `  U )   &    |- 
 .h  =  ( .s
 OLD `  U )   &    |-  .ih  =  ( .i OLD `  U )   &    |-  U  e.  NrmCVec   =>    |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.
 
Theoremhhnv 21574 Hilbert space is a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   =>    |-  U  e.  NrmCVec
 
Theoremhhva 21575 The group (addition) operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   =>    |- 
 +h  =  ( +v
 `  U )
 
Theoremhhba 21576 The base set of Hilbert space. This theorem provides an independent proof of df-hba 21379 (see comments in that definition). (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   =>    |- 
 ~H  =  ( BaseSet `  U )
 
Theoremhh0v 21577 The zero vector of Hilbert space. (Contributed by NM, 17-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   =>    |- 
 0h  =  ( 0vec `  U )
 
Theoremhhsm 21578 The scalar product operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   =>    |- 
 .h  =  ( .s
 OLD `  U )
 
Theoremhhvs 21579 The vector subtraction operation of Hilbert space. (Contributed by NM, 13-Dec-2007.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   =>    |- 
 -h  =  ( -v
 `  U )
 
Theoremhhnm 21580 The norm function of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   =>    |- 
 normh  =  ( normCV `  U )
 
Theoremhhims 21581 The induced metric of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  D  =  ( normh  o. 
 -h  )   =>    |-  D  =  ( IndMet `  U )
 
Theoremhhims2 21582 Hilbert space distance metric. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  D  =  ( IndMet `  U )   =>    |-  D  =  ( normh  o. 
 -h  )
 
Theoremhhmet 21583 The induced metric of Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  D  =  ( IndMet `  U )   =>    |-  D  e.  ( Met `  ~H )
 
Theoremhhxmet 21584 The induced metric of Hilbert space. (Contributed by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  D  =  ( IndMet `  U )   =>    |-  D  e.  ( * Met `  ~H )
 
Theoremhhmetdval 21585 Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   &    |-  D  =  ( IndMet `  U )   =>    |-  ( ( A  e.  ~H 
 /\  B  e.  ~H )  ->  ( A D B )  =  ( normh `  ( A  -h  B ) ) )
 
Theoremhhip 21586 The inner product operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   =>    |- 
 .ih  =  ( .i OLD `  U )
 
Theoremhhph 21587 The Hilbert space of the Hilbert Space Explorer is an inner product space. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
 |-  U  =  <. <.  +h  ,  .h  >. ,  normh >.   =>    |-  U  e.  CPreHil OLD
 
15.9.10  Bunjakovaskij-Cauchy-Schwarz inequality
 
TheorembcsiALT 21588 Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( abs `  ( A  .ih  B ) )  <_  (
 ( normh `  A )  x.  ( normh `  B )
 )
 
TheorembcsiHIL 21589 Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Proved from ZFC version.) (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.)
 |-  A  e.  ~H   &    |-  B  e.  ~H   =>    |-  ( abs `  ( A  .ih  B ) )  <_  (
 ( normh `  A )  x.  ( normh `  B )
 )
 
Theorembcs 21590 Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H )  ->  ( abs `  ( A  .ih  B ) ) 
 <_  ( ( normh `  A )  x.  ( normh `  B ) ) )
 
Theorembcs2 21591 Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 21589. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  ( normh `  A )  <_  1 )  ->  ( abs `  ( A  .ih  B ) )  <_  ( normh `  B ) )
 
Theorembcs3 21592 Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 21589. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
 |-  (
 ( A  e.  ~H  /\  B  e.  ~H  /\  ( normh `  B )  <_  1 )  ->  ( abs `  ( A  .ih  B ) )  <_  ( normh `  A ) )
 
15.9.11  Cauchy sequences and limits
 
Theoremhcau 21593* Member of the set of Cauchy sequences on a Hilbert space. Definition for Cauchy sequence in [Beran] p. 96. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ( F  e.  Cauchy  <->  ( F : NN
 --> ~H  /\  A. x  e.  RR+  E. y  e. 
 NN  A. z  e.  ( ZZ>=
 `  y ) (
 normh `  ( ( F `
  y )  -h  ( F `  z ) ) )  <  x ) )
 
Theoremhcauseq 21594 A Cauchy sequences on a Hilbert space is a sequence. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ( F  e.  Cauchy  ->  F : NN --> ~H )
 
Theoremhcaucvg 21595* A Cauchy sequence on a Hilbert space converges. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  (
 ( F  e.  Cauchy  /\  A  e.  RR+ )  ->  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  y )  -h  ( F `  z ) ) )  <  A )
 
Theoremseq1hcau 21596* A sequence on a Hilbert space is a Cauchy sequence if it converges. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  ( F : NN --> ~H  ->  ( F  e.  Cauchy  <->  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  y )  -h  ( F `  z ) ) )  <  x ) )
 
Theoremhlimi 21597* Express the predicate: The limit of vector sequence  F in a Hilbert space is  A, i.e.  F converges to  A. This means that for any real  x, no matter how small, there always exists an integer  y such that the norm of any later vector in the sequence minus the limit is less than  x. Definition of converge in [Beran] p. 96. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( F  ~~>v  A  <->  ( ( F : NN --> ~H  /\  A  e.  ~H )  /\  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  z )  -h  A ) )  <  x ) )
 
Theoremhlimseqi 21598 A sequence with a limit on a Hilbert space is a sequence. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( F  ~~>v  A  ->  F : NN --> ~H )
 
Theoremhlimveci 21599 Closure of the limit of a sequence on Hilbert space. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( F  ~~>v  A  ->  A  e.  ~H )
 
Theoremhlimconvi 21600* Convergence of a sequence on a Hilbert space. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( ( F  ~~>v  A 
 /\  B  e.  RR+ )  ->  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  z )  -h  A ) )  <  B )
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