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Theorem List for Metamath Proof Explorer - 22501-22600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremadjeq0 22501 An operator is zero iff its adjoint is zero. Theorem 3.11(i) of [Beran] p. 106. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)

Theoremadjmul 22502 The adjoint of the scalar product of an operator. Theorem 3.11(ii) of [Beran] p. 106. (Contributed by NM, 21-Feb-2006.) (New usage is discouraged.)

Theoremadjadd 22503 The adjoint of the sum of two operators. Theorem 3.11(iii) of [Beran] p. 106. (Contributed by NM, 22-Feb-2006.) (New usage is discouraged.)

Theoremnmoptrii 22504 Triangle inequality for the norms of bounded linear operators. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)

Theoremnmopcoi 22505 Upper bound for the norm of the composition of two bounded linear operators. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)

Theorembdophsi 22506 The sum of two bounded linear operators is a bounded linear operator. (Contributed by NM, 9-Mar-2006.) (New usage is discouraged.)

Theorembdophdi 22507 The difference between two bounded linear operators is bounded. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)

Theorembdopcoi 22508 The composition of two bounded linear operators is bounded. (Contributed by NM, 9-Mar-2006.) (New usage is discouraged.)

Theoremnmoptri2i 22509 Triangle-type inequality for the norms of bounded linear operators. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)

Theoremadjcoi 22510 The adjoint of a composition of bounded linear operators. Theorem 3.11(viii) of [Beran] p. 106. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)

Theoremnmopcoadji 22511 The norm of an operator composed with its adjoint. Part of Theorem 3.11(vi) of [Beran] p. 106. (Contributed by NM, 8-Mar-2006.) (New usage is discouraged.)

Theoremnmopcoadj2i 22512 The norm of an operator composed with its adjoint. Part of Theorem 3.11(vi) of [Beran] p. 106. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)

Theoremnmopcoadj0i 22513 An operator composed with its adjoint is zero iff the operator is zero. Theorem 3.11(vii) of [Beran] p. 106. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)

15.9.44  Quantum computation error bound theorem

Theoremunierri 22514 If we approximate a chain of unitary transformations (quantum computer gates) , by other unitary transformations , , the error increases at most additively. Equation 4.73 of [NielsenChuang] p. 195. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)

15.9.45  Dirac bra-ket notation (cont.)

Theorembranmfn 22515 The norm of the bra function. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)

Theorembrabn 22516 The bra of a vector is a bounded functional. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)

Theoremrnbra 22517 The set of bras equals the set of continuous linear functionals. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)

Theorembra11 22518 The bra function maps vectors one-to-one onto the set of continuous linear functionals. (Contributed by NM, 26-May-2006.) (Proof shortened by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theorembracnln 22519 A bra is a continuous linear functional. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)

Theoremcnvbraval 22520* Value of the converse of the bra function. Based on the Riesz Lemma riesz4 22474, this very important theorem not only justifies the Dirac bra-ket notation, but allows us to extract a unique vector from any continuous linear functional from which the functional can be recovered; i.e. a single vector can "store" all of the information contained in any entire continuous linear functional (mapping from to ). (Contributed by NM, 26-May-2006.) (New usage is discouraged.)

Theoremcnvbracl 22521 Closure of the converse of the bra function. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)

Theoremcnvbrabra 22522 The converse bra of the bra of a vector is the vector itself. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)

Theorembracnvbra 22523 The bra of the converse bra of a continuous linear functional. (Contributed by NM, 31-May-2006.) (New usage is discouraged.)

Theorembracnlnval 22524* The vector that a continuous linear functional is the bra of. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)

Theoremcnvbramul 22525 Multiplication property of the converse bra function. (Contributed by NM, 31-May-2006.) (New usage is discouraged.)

Theoremkbass1 22526 Dirac bra-ket associative law i.e. the juxtaposition of an outer product with a ket equals a bra juxtaposed with an inner product. Since is a complex number, it is the first argument in the inner product that it is mapped to, although in Dirac notation it is placed after the ket. (Contributed by NM, 15-May-2006.) (New usage is discouraged.)

Theoremkbass2 22527 Dirac bra-ket associative law i.e. the juxtaposition of an inner product with a bra equals a ket juxtaposed with an outer product. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)

Theoremkbass3 22528 Dirac bra-ket associative law . (Contributed by NM, 30-May-2006.) (New usage is discouraged.)

Theoremkbass4 22529 Dirac bra-ket associative law . (Contributed by NM, 30-May-2006.) (New usage is discouraged.)

Theoremkbass5 22530 Dirac bra-ket associative law . (Contributed by NM, 30-May-2006.) (New usage is discouraged.)

Theoremkbass6 22531 Dirac bra-ket associative law . (Contributed by NM, 30-May-2006.) (New usage is discouraged.)

15.9.46  Positive operators (cont.)

Theoremleopg 22532* 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 22533* Ordering relation for operators. Definition of positive operator ordering in [Kreyszig] p. 470. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)

Theoremleop2 22534* Ordering relation for operators. Definition of operator ordering in [Young] p. 141. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)

Theoremleop3 22535 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 22536* Binary relation defining a positive operator. Definition VI.1 of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)

Theoremleoprf2 22537 The ordering relation for operators is reflexive. (Contributed by NM, 24-Jul-2006.) (New usage is discouraged.)

Theoremleoprf 22538 The ordering relation for operators is reflexive. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)

Theoremleopsq 22539 The square of a Hermitian operator is positive. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)

Theorem0leop 22540 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 22541 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 22542 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 22543 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 22544 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 22545 Scalar product applied to operator ordering. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)

Theoremleoptri 22546 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 22547 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 22548 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 22549 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 22550* Lemma for opsqri . (Contributed by NM, 9-Aug-2006.) (New usage is discouraged.)

Theoremopsqrlem2 22551* 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 22552* Lemma for opsqri . (Contributed by NM, 22-Aug-2006.) (New usage is discouraged.)

Theoremopsqrlem4 22553* Lemma for opsqri . (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)

Theoremopsqrlem5 22554* Lemma for opsqri . (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)

Theoremopsqrlem6 22555* Lemma for opsqri . (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)

15.9.47  Projectors as operators

Theorempjhmopi 22556 A projector is a Hermitian operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)

Theorempjlnopi 22557 A projector is a linear operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)

Theorempjnmopi 22558 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 22559 A projector is a bounded linear operator. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)

Theorempjhmop 22560 A projection is a Hermitian operator. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theoremhmopidmchi 22561 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 22562 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 22561 shows.) (Contributed by NM, 22-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)

Theoremhmopidmch 22563 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 22564 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 22565 Distributive law for Hilbert space operator sum. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)

Theorempjddii 22566 Distributive law for Hilbert space operator difference. (Contributed by NM, 24-Nov-2000.) (New usage is discouraged.)

Theorempjsdi2i 22567 Chained distributive law for Hilbert space operator difference. (Contributed by NM, 30-Nov-2000.) (New usage is discouraged.)

Theorempjcoi 22568 Composition of projections. (Contributed by NM, 16-Aug-2000.) (New usage is discouraged.)

Theorempjcocli 22569 Closure of composition of projections. (Contributed by NM, 29-Nov-2000.) (New usage is discouraged.)

Theorempjcohcli 22570 Closure of composition of projections. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)

Theorempjadjcoi 22571 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 22572 Functionality of composition of projections. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)

Theorempjss1coi 22573 Subset relationship for projections. Theorem 4.5(i)<->(iii) of [Beran] p. 112. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)

Theorempjss2coi 22574 Subset relationship for projections. Theorem 4.5(i)<->(ii) of [Beran] p. 112. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)

Theorempjssmi 22575 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 22576 Theorem 4.5(iv)->(v) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)

Theorempjdifnormi 22577 Theorem 4.5(v)<->(vi) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)

Theorempjnormssi 22578* Theorem 4.5(i)<->(vi) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)

Theorempjorthcoi 22579 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 22580 The projection of orthogonal subspaces is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)

Theorempjssumi 22581 The projection on a subspace sum is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)

Theorempjssposi 22582* 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 22583* 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 22582). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)

Theorempjssdif2i 22584 The projection subspace of the difference between two projectors. Part 2 of Theorem 29.3 of [Halmos] p. 48 (shortened with pjssposi 22582). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)

Theorempjssdif1i 22585 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 22582). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)

Theorempjimai 22586 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 22587 A projection is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)

Theorempjoccoi 22588 Composition of projections of a subspace and its orthocomplement. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)

Theorempjtoi 22589 Subspace sum of projection and projection of orthocomplement. (Contributed by NM, 16-Nov-2000.) (New usage is discouraged.)

Theorempjoci 22590 Projection of orthocomplement. First part of Theorem 27.3 of [Halmos] p. 45. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Theorempjidmco 22591 A projection operator is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theoremdfpjop 22592 Definition of projection operator in [Hughes] p. 47, except that we do not need linearity to be explicit by virtue of hmoplin 22352. (Contributed by NM, 24-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)

Theorempjhmopidm 22593 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 22594 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 22595 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 22596 A projector is a positive operator. (Contributed by NM, 27-Sep-2008.) (New usage is discouraged.)

Theorempjadj2 22597 A projector is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)

Theorempjadj3 22598 A projector is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)

Theoremelpjch 22599 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 22600* 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.)

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