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Theorem List for Metamath Proof Explorer - 22701-22800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremkbass6 22701 Dirac bra-ket associative law . (Contributed by NM, 30-May-2006.) (New usage is discouraged.)

17.6.15  Positive operators (cont.)

Theoremleopg 22702* Ordering relation for positive operators. Definition of positive operator ordering in [Kreyszig] p. 470. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)

Theoremleop 22703* Ordering relation for operators. Definition of positive operator ordering in [Kreyszig] p. 470. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)

Theoremleop2 22704* Ordering relation for operators. Definition of operator ordering in [Young] p. 141. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)

Theoremleop3 22705 Operator ordering in terms of a positive operator. Definition of operator ordering in [Retherford] p. 49. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)

Theoremleoppos 22706* Binary relation defining a positive operator. Definition VI.1 of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)

Theoremleoprf2 22707 The ordering relation for operators is reflexive. (Contributed by NM, 24-Jul-2006.) (New usage is discouraged.)

Theoremleoprf 22708 The ordering relation for operators is reflexive. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)

Theoremleopsq 22709 The square of a Hermitian operator is positive. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)

Theorem0leop 22710 The zero operator is a positive operator. (The literature calls it "positive," even though in some sense it is really "nonnegative.") Part of Example 12.2(i) in [Young] p. 142. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)

Theoremidleop 22711 The identity operator is a positive operator. Part of Example 12.2(i) in [Young] p. 142. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)

Theoremleopadd 22712 The sum of two positive operators is positive. Exercise 1(i) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)

Theoremleopmuli 22713 The scalar product of a nonnegative real and a positive operator is a positive operator. Exercise 1(ii) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)

Theoremleopmul 22714 The scalar product of a positive real and a positive operator is a positive operator. Exercise 1(ii) of [Retherford] p. 49. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)

Theoremleopmul2i 22715 Scalar product applied to operator ordering. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)

Theoremleoptri 22716 The positive operator ordering relation satisfies trichotomy. Exercise 1(iii) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)

Theoremleoptr 22717 The positive operator ordering relation is transitive. Exercise 1(iv) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)

Theoremleopnmid 22718 A bounded Hermitian operator is less than or equal to its norm times the identity operator. (Contributed by NM, 11-Aug-2006.) (New usage is discouraged.)

Theoremnmopleid 22719 A nonzero, bounded Hermitian operator divided by its norm is less than or equal to the identity operator. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)

Theoremopsqrlem1 22720* Lemma for opsqri . (Contributed by NM, 9-Aug-2006.) (New usage is discouraged.)

Theoremopsqrlem2 22721* Lemma for opsqri . is the recursive function An (starting at n=1 instead of 0) of Theorem 9.4-2 of [Kreyszig] p. 476. (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)

Theoremopsqrlem3 22722* Lemma for opsqri . (Contributed by NM, 22-Aug-2006.) (New usage is discouraged.)

Theoremopsqrlem4 22723* Lemma for opsqri . (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)

Theoremopsqrlem5 22724* Lemma for opsqri . (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)

Theoremopsqrlem6 22725* Lemma for opsqri . (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)

17.6.16  Projectors as operators

Theorempjhmopi 22726 A projector is a Hermitian operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)

Theorempjlnopi 22727 A projector is a linear operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)

Theorempjnmopi 22728 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 22729 A projector is a bounded linear operator. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)

Theorempjhmop 22730 A projection is a Hermitian operator. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theoremhmopidmchi 22731 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 22732 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 22731 shows.) (Contributed by NM, 22-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)

Theoremhmopidmch 22733 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 22734 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 22735 Distributive law for Hilbert space operator sum. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)

Theorempjddii 22736 Distributive law for Hilbert space operator difference. (Contributed by NM, 24-Nov-2000.) (New usage is discouraged.)

Theorempjsdi2i 22737 Chained distributive law for Hilbert space operator difference. (Contributed by NM, 30-Nov-2000.) (New usage is discouraged.)

Theorempjcoi 22738 Composition of projections. (Contributed by NM, 16-Aug-2000.) (New usage is discouraged.)

Theorempjcocli 22739 Closure of composition of projections. (Contributed by NM, 29-Nov-2000.) (New usage is discouraged.)

Theorempjcohcli 22740 Closure of composition of projections. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)

Theorempjadjcoi 22741 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 22742 Functionality of composition of projections. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)

Theorempjss1coi 22743 Subset relationship for projections. Theorem 4.5(i)<->(iii) of [Beran] p. 112. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)

Theorempjss2coi 22744 Subset relationship for projections. Theorem 4.5(i)<->(ii) of [Beran] p. 112. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)

Theorempjssmi 22745 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 22746 Theorem 4.5(iv)->(v) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)

Theorempjdifnormi 22747 Theorem 4.5(v)<->(vi) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)

Theorempjnormssi 22748* Theorem 4.5(i)<->(vi) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)

Theorempjorthcoi 22749 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 22750 The projection of orthogonal subspaces is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)

Theorempjssumi 22751 The projection on a subspace sum is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)

Theorempjssposi 22752* 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 22753* 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 22752). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)

Theorempjssdif2i 22754 The projection subspace of the difference between two projectors. Part 2 of Theorem 29.3 of [Halmos] p. 48 (shortened with pjssposi 22752). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)

Theorempjssdif1i 22755 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 22752). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)

Theorempjimai 22756 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 22757 A projection is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)

Theorempjoccoi 22758 Composition of projections of a subspace and its orthocomplement. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)

Theorempjtoi 22759 Subspace sum of projection and projection of orthocomplement. (Contributed by NM, 16-Nov-2000.) (New usage is discouraged.)

Theorempjoci 22760 Projection of orthocomplement. First part of Theorem 27.3 of [Halmos] p. 45. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Theorempjidmco 22761 A projection operator is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theoremdfpjop 22762 Definition of projection operator in [Hughes] p. 47, except that we do not need linearity to be explicit by virtue of hmoplin 22522. (Contributed by NM, 24-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)

Theorempjhmopidm 22763 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 22764 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 22765 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 22766 A projector is a positive operator. (Contributed by NM, 27-Sep-2008.) (New usage is discouraged.)

Theorempjadj2 22767 A projector is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)

Theorempjadj3 22768 A projector is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)

Theoremelpjch 22769 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 22770* 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 22771 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 22772 Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)

Theorempjin2i 22773 Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)

Theorempjin3i 22774 Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)

Theorempjclem1 22775 Lemma for projection commutation theorem. (Contributed by NM, 16-Nov-2000.) (New usage is discouraged.)

Theorempjclem2 22776 Lemma for projection commutation theorem. (Contributed by NM, 17-Nov-2000.) (New usage is discouraged.)

Theorempjclem3 22777 Lemma for projection commutation theorem. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Theorempjclem4a 22778 Lemma for projection commutation theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)

Theorempjclem4 22779 Lemma for projection commutation theorem. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Theorempjci 22780 Two subspaces commute iff their projections commute. Lemma 4 of [Kalmbach] p. 67. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Theorempjcmul1i 22781 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 22782 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 22783 Closure of composition of projection and Hilbert space operator. (Contributed by NM, 3-Dec-2000.) (New usage is discouraged.)

Theorempjadj2coi 22784 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 22785 Closure of double composition of projections. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)

Theorempj3lem1 22786 Lemma for projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)

Theorempj3si 22787 Stronger projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)

Theorempj3i 22788 Projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)

Theorempj3cor1i 22789 Projection triplet corollary. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.)

Theorempjs14i 22790 Theorem S-14 of Watanabe, p. 486. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)

17.7  States on an Hilbert lattice and Godowski's equation

17.7.1  States on a Hilbert lattice

Definitiondf-st 22791* 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 22792* 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 22793* Property of a state. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremishst 22794* 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 22795 closure of the value of a state. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremstcl 22796 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 22797 Closure of the value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhst1a 22798 Unit value of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhstel2 22799 Properties of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

Theoremhstorth 22800 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.)

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