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Theorem List for Metamath Proof Explorer - 23601-23700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremopsqrlem6 23601* Lemma for opsqri . (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)

18.6.16  Projectors as operators

Theorempjhmopi 23602 A projector is a Hermitian operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)

Theorempjlnopi 23603 A projector is a linear operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)

Theorempjnmopi 23604 The operator norm of a projector on a nonzero closed subspace is one. Part of Theorem 26.1 of [Halmos] p. 43. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.)

Theorempjbdlni 23605 A projector is a bounded linear operator. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)

Theorempjhmop 23606 A projection is a Hermitian operator. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theoremhmopidmchi 23607 An idempotent Hermitian operator generates a closed subspace. Part of proof of Theorem of [AkhiezerGlazman] p. 64. (Contributed by NM, 21-Apr-2006.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)

Theoremhmopidmpji 23608 An idempotent Hermitian operator is a projection operator. Theorem 26.4 of [Halmos] p. 44. (Halmos seems to omit the proof that is a closed subspace, which is not trivial as hmopidmchi 23607 shows.) (Contributed by NM, 22-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)

Theoremhmopidmch 23609 An idempotent Hermitian operator generates a closed subspace. Part of proof of Theorem of [AkhiezerGlazman] p. 64. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theoremhmopidmpj 23610 An idempotent Hermitian operator is a projection operator. Theorem 26.4 of [Halmos] p. 44. (Contributed by NM, 22-Apr-2006.) (New usage is discouraged.)

Theorempjsdii 23611 Distributive law for Hilbert space operator sum. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)

Theorempjddii 23612 Distributive law for Hilbert space operator difference. (Contributed by NM, 24-Nov-2000.) (New usage is discouraged.)

Theorempjsdi2i 23613 Chained distributive law for Hilbert space operator difference. (Contributed by NM, 30-Nov-2000.) (New usage is discouraged.)

Theorempjcoi 23614 Composition of projections. (Contributed by NM, 16-Aug-2000.) (New usage is discouraged.)

Theorempjcocli 23615 Closure of composition of projections. (Contributed by NM, 29-Nov-2000.) (New usage is discouraged.)

Theorempjcohcli 23616 Closure of composition of projections. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)

Theorempjadjcoi 23617 Adjoint of composition of projections. Special case of Theorem 3.11(viii) of [Beran] p. 106. (Contributed by NM, 6-Oct-2000.) (New usage is discouraged.)

Theorempjcofni 23618 Functionality of composition of projections. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)

Theorempjss1coi 23619 Subset relationship for projections. Theorem 4.5(i)<->(iii) of [Beran] p. 112. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)

Theorempjss2coi 23620 Subset relationship for projections. Theorem 4.5(i)<->(ii) of [Beran] p. 112. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)

Theorempjssmi 23621 Projection meet property. Remark in [Kalmbach] p. 66. Also Theorem 4.5(i)->(iv) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)

Theorempjssge0i 23622 Theorem 4.5(iv)->(v) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)

Theorempjdifnormi 23623 Theorem 4.5(v)<->(vi) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)

Theorempjnormssi 23624* Theorem 4.5(i)<->(vi) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)

Theorempjorthcoi 23625 Composition of projections of orthogonal subspaces. Part (i)->(iia) of Theorem 27.4 of [Halmos] p. 45. (Contributed by NM, 6-Nov-2000.) (New usage is discouraged.)

Theorempjscji 23626 The projection of orthogonal subspaces is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)

Theorempjssumi 23627 The projection on a subspace sum is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)

Theorempjssposi 23628* Projector ordering can be expressed by the subset relationship between their projection subspaces. (i)<->(iii) of Theorem 29.2 of [Halmos] p. 48. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)

Theorempjordi 23629* The definition of projector ordering in [Halmos] p. 42 is equivalent to the definition of projector ordering in [Beran] p. 110. (We will usually express projector ordering with the even simpler equivalent ; see pjssposi 23628). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)

Theorempjssdif2i 23630 The projection subspace of the difference between two projectors. Part 2 of Theorem 29.3 of [Halmos] p. 48 (shortened with pjssposi 23628). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)

Theorempjssdif1i 23631 A necessary and sufficient condition for the difference between two projectors to be a projector. Part 1 of Theorem 29.3 of [Halmos] p. 48 (shortened with pjssposi 23628). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)

Theorempjimai 23632 The image of a projection. Lemma 5 in Daniel Lehmann, "A presentation of Quantum Logic based on an and then connective" http://www.arxiv.org/pdf/quant-ph/0701113 p. 20. (Contributed by NM, 20-Jan-2007.) (New usage is discouraged.)

Theorempjidmcoi 23633 A projection is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)

Theorempjoccoi 23634 Composition of projections of a subspace and its orthocomplement. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)

Theorempjtoi 23635 Subspace sum of projection and projection of orthocomplement. (Contributed by NM, 16-Nov-2000.) (New usage is discouraged.)

Theorempjoci 23636 Projection of orthocomplement. First part of Theorem 27.3 of [Halmos] p. 45. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Theorempjidmco 23637 A projection operator is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theoremdfpjop 23638 Definition of projection operator in [Hughes] p. 47, except that we do not need linearity to be explicit by virtue of hmoplin 23398. (Contributed by NM, 24-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)

Theorempjhmopidm 23639 Two ways to express the set of all projection operators. (Contributed by NM, 24-Apr-2006.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)

Theoremelpjidm 23640 A projection operator is idempotent. Part of Theorem 26.1 of [Halmos] p. 43. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theoremelpjhmop 23641 A projection operator is Hermitian. Part of Theorem 26.1 of [Halmos] p. 43. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theorem0leopj 23642 A projector is a positive operator. (Contributed by NM, 27-Sep-2008.) (New usage is discouraged.)

Theorempjadj2 23643 A projector is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)

Theorempjadj3 23644 A projector is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)

Theoremelpjch 23645 Reconstruction of the subspace of a projection operator. Part of Theorem 26.2 of [Halmos] p. 44. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theoremelpjrn 23646* Reconstruction of the subspace of a projection operator. (Contributed by NM, 24-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)

Theorempjinvari 23647 A closed subspace with projection is invariant under an operator iff . Theorem 27.1 of [Halmos] p. 45. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theorempjin1i 23648 Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)

Theorempjin2i 23649 Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)

Theorempjin3i 23650 Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)

Theorempjclem1 23651 Lemma for projection commutation theorem. (Contributed by NM, 16-Nov-2000.) (New usage is discouraged.)

Theorempjclem2 23652 Lemma for projection commutation theorem. (Contributed by NM, 17-Nov-2000.) (New usage is discouraged.)

Theorempjclem3 23653 Lemma for projection commutation theorem. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Theorempjclem4a 23654 Lemma for projection commutation theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)

Theorempjclem4 23655 Lemma for projection commutation theorem. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Theorempjci 23656 Two subspaces commute iff their projections commute. Lemma 4 of [Kalmbach] p. 67. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Theorempjcmul1i 23657 A necessary and sufficient condition for the product of two projectors to be a projector is that the projectors commute. Part 1 of Theorem 1 of [AkhiezerGlazman] p. 65. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)

Theorempjcmul2i 23658 The projection subspace of the difference between two projectors. Part 2 of Theorem 1 of [AkhiezerGlazman] p. 65. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)

Theorempjcohocli 23659 Closure of composition of projection and Hilbert space operator. (Contributed by NM, 3-Dec-2000.) (New usage is discouraged.)

Theorempjadj2coi 23660 Adjoint of double composition of projections. Generalization of special case of Theorem 3.11(viii) of [Beran] p. 106. (Contributed by NM, 1-Dec-2000.) (New usage is discouraged.)

Theorempj2cocli 23661 Closure of double composition of projections. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)

Theorempj3lem1 23662 Lemma for projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)

Theorempj3si 23663 Stronger projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)

Theorempj3i 23664 Projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)

Theorempj3cor1i 23665 Projection triplet corollary. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)

Theorempjs14i 23666 Theorem S-14 of Watanabe, p. 486. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)

18.7  States on a Hilbert lattice and Godowski's equation

18.7.1  States on a Hilbert lattice

Definitiondf-st 23667* Define the set of states on a Hilbert lattice. Definition of [Kalmbach] p. 266. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)

Definitiondf-hst 23668* Define the set of complex Hilbert-space-valued states on a Hilbert lattice. Definition of CH-states in [Mayet3] p. 9. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremisst 23669* Property of a state. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremishst 23670* Property of a complex Hilbert-space-valued state. Definition of CH-states in [Mayet3] p. 9. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremsticl 23671 closure of the value of a state. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremstcl 23672 Real closure of the value of a state. (Contributed by NM, 24-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremhstcl 23673 Closure of the value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhst1a 23674 Unit value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhstel2 23675 Properties of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhstorth 23676 Orthogonality property of a Hilbert-space-valued state. This is a key feature distinguishing it from a real-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhstosum 23677 Orthogonal sum property of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhstoc 23678 Sum of a Hilbert-space-valued state of a lattice element and its orthocomplement. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhstnmoc 23679 Sum of norms of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremstge0 23680 The value of a state is nonnegative. (Contributed by NM, 24-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremstle1 23681 The value of a state is less than or equal to one. (Contributed by NM, 24-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremhstle1 23682 The norm of the value of a Hilbert-space-valued state is less than or equal to one. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhst1h 23683 The norm of a Hilbert-space-valued state equals one iff the state value equals the state value of the lattice unit. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhst0h 23684 The norm of a Hilbert-space-valued state equals zero iff the state value equals zero. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremhstpyth 23685 Pythagorean property of a Hilbert-space-valued state for orthogonal vectors and . (Contributed by NM, 26-Jun-2006.) (New usage is discouraged.)

Theoremhstle 23686 Ordering property of a Hilbert-space-valued state. (Contributed by NM, 26-Jun-2006.) (New usage is discouraged.)

Theoremhstles 23687 Ordering property of a Hilbert-space-valued state. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremhstoh 23688 A Hilbert-space-valued state orthogonal to the state of the lattice unit is zero. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhst0 23689 A Hilbert-space-valued state is zero at the zero subspace. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theoremsthil 23690 The value of a state at the full Hilbert space. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)

Theoremstj 23691 The value of a state on a join. (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)

Theoremsto1i 23692 The state of a subspace plus the state of its orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremsto2i 23693 The state of the orthocomplement. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstge1i 23694 If a state is greater than or equal to 1, it is 1. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)

Theoremstle0i 23695 If a state is less than or equal to 0, it is 0. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)

Theoremstlei 23696 Ordering law for states. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstlesi 23697 Ordering law for states. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstji1i 23698 Join of components of Sasaki arrow ->1. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremstm1i 23699 State of component of unit meet. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)

Theoremstm1ri 23700 State of component of unit meet. (Contributed by NM, 11-Nov-1999.) (New usage is discouraged.)

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