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Theorem List for Metamath Proof Explorer - 29801-29900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembdophsi 29801 The sum of two bounded linear operators is a bounded linear operator. (Contributed by NM, 9-Mar-2006.) (New usage is discouraged.)
𝑆 ∈ BndLinOp    &   𝑇 ∈ BndLinOp       (𝑆 +op 𝑇) ∈ BndLinOp
 
Theorembdophdi 29802 The difference between two bounded linear operators is bounded. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑆 ∈ BndLinOp    &   𝑇 ∈ BndLinOp       (𝑆op 𝑇) ∈ BndLinOp
 
Theorembdopcoi 29803 The composition of two bounded linear operators is bounded. (Contributed by NM, 9-Mar-2006.) (New usage is discouraged.)
𝑆 ∈ BndLinOp    &   𝑇 ∈ BndLinOp       (𝑆𝑇) ∈ BndLinOp
 
Theoremnmoptri2i 29804 Triangle-type inequality for the norms of bounded linear operators. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝑆 ∈ BndLinOp    &   𝑇 ∈ BndLinOp       ((normop𝑆) − (normop𝑇)) ≤ (normop‘(𝑆 +op 𝑇))
 
Theoremadjcoi 29805 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.)
𝑆 ∈ BndLinOp    &   𝑇 ∈ BndLinOp       (adj‘(𝑆𝑇)) = ((adj𝑇) ∘ (adj𝑆))
 
Theoremnmopcoadji 29806 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.)
𝑇 ∈ BndLinOp       (normop‘((adj𝑇) ∘ 𝑇)) = ((normop𝑇)↑2)
 
Theoremnmopcoadj2i 29807 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.)
𝑇 ∈ BndLinOp       (normop‘(𝑇 ∘ (adj𝑇))) = ((normop𝑇)↑2)
 
Theoremnmopcoadj0i 29808 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.)
𝑇 ∈ BndLinOp       ((𝑇 ∘ (adj𝑇)) = 0hop𝑇 = 0hop )
 
19.6.13  Quantum computation error bound theorem
 
Theoremunierri 29809 If we approximate a chain of unitary transformations (quantum computer gates) 𝐹, 𝐺 by other unitary transformations 𝑆, 𝑇, the error increases at most additively. Equation 4.73 of [NielsenChuang] p. 195. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)
𝐹 ∈ UniOp    &   𝐺 ∈ UniOp    &   𝑆 ∈ UniOp    &   𝑇 ∈ UniOp       (normop‘((𝐹𝐺) −op (𝑆𝑇))) ≤ ((normop‘(𝐹op 𝑆)) + (normop‘(𝐺op 𝑇)))
 
19.6.14  Dirac bra-ket notation (cont.)
 
Theorembranmfn 29810 The norm of the bra function. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) = (norm𝐴))
 
Theorembrabn 29811 The bra of a vector is a bounded functional. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (normfn‘(bra‘𝐴)) ∈ ℝ)
 
Theoremrnbra 29812 The set of bras equals the set of continuous linear functionals. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
ran bra = (LinFn ∩ ContFn)
 
Theorembra11 29813 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.)
bra: ℋ–1-1-onto→(LinFn ∩ ContFn)
 
Theorembracnln 29814 A bra is a continuous linear functional. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (bra‘𝐴) ∈ (LinFn ∩ ContFn))
 
Theoremcnvbraval 29815* Value of the converse of the bra function. Based on the Riesz Lemma riesz4 29769, 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.)
(𝑇 ∈ (LinFn ∩ ContFn) → (bra‘𝑇) = (𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇𝑥) = (𝑥 ·ih 𝑦)))
 
Theoremcnvbracl 29816 Closure of the converse of the bra function. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
(𝑇 ∈ (LinFn ∩ ContFn) → (bra‘𝑇) ∈ ℋ)
 
Theoremcnvbrabra 29817 The converse bra of the bra of a vector is the vector itself. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
(𝐴 ∈ ℋ → (bra‘(bra‘𝐴)) = 𝐴)
 
Theorembracnvbra 29818 The bra of the converse bra of a continuous linear functional. (Contributed by NM, 31-May-2006.) (New usage is discouraged.)
(𝑇 ∈ (LinFn ∩ ContFn) → (bra‘(bra‘𝑇)) = 𝑇)
 
Theorembracnlnval 29819* The vector that a continuous linear functional is the bra of. (Contributed by NM, 26-May-2006.) (New usage is discouraged.)
(𝑇 ∈ (LinFn ∩ ContFn) → 𝑇 = (bra‘(𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑇𝑥) = (𝑥 ·ih 𝑦))))
 
Theoremcnvbramul 29820 Multiplication property of the converse bra function. (Contributed by NM, 31-May-2006.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝑇 ∈ (LinFn ∩ ContFn)) → (bra‘(𝐴 ·fn 𝑇)) = ((∗‘𝐴) · (bra‘𝑇)))
 
Theoremkbass1 29821 Dirac bra-ket associative law ( ∣ 𝐴 𝐵 ∣ ) ∣ 𝐶⟩ = 𝐴⟩(⟨𝐵𝐶⟩) i.e. the juxtaposition of an outer product with a ket equals a bra juxtaposed with an inner product. Since 𝐵𝐶 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.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = (((bra‘𝐵)‘𝐶) · 𝐴))
 
Theoremkbass2 29822 Dirac bra-ket associative law (⟨𝐴𝐵⟩)⟨𝐶 ∣ = 𝐴 ∣ ( ∣ 𝐵 𝐶 ∣ ) 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.)
((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (((bra‘𝐴)‘𝐵) ·fn (bra‘𝐶)) = ((bra‘𝐴) ∘ (𝐵 ketbra 𝐶)))
 
Theoremkbass3 29823 Dirac bra-ket associative law 𝐴𝐵 𝐶𝐷⟩ = (⟨𝐴𝐵 𝐶 ∣ ) ∣ 𝐷. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
(((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((bra‘𝐴)‘𝐵) · ((bra‘𝐶)‘𝐷)) = ((((bra‘𝐴)‘𝐵) ·fn (bra‘𝐶))‘𝐷))
 
Theoremkbass4 29824 Dirac bra-ket associative law 𝐴𝐵 𝐶𝐷⟩ = 𝐴 ∣ ( ∣ 𝐵 𝐶𝐷⟩). (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
(((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((bra‘𝐴)‘𝐵) · ((bra‘𝐶)‘𝐷)) = ((bra‘𝐴)‘(((bra‘𝐶)‘𝐷) · 𝐵)))
 
Theoremkbass5 29825 Dirac bra-ket associative law ( ∣ 𝐴 𝐵 ∣ )( ∣ 𝐶 𝐷 ∣ ) = (( ∣ 𝐴 𝐵 ∣ ) ∣ 𝐶⟩)⟨𝐷. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
(((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)) = (((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷))
 
Theoremkbass6 29826 Dirac bra-ket associative law ( ∣ 𝐴 𝐵 ∣ )( ∣ 𝐶 𝐷 ∣ ) = ∣ 𝐴 (⟨𝐵 ∣ ( ∣ 𝐶 𝐷 ∣ )). (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
(((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)) = (𝐴 ketbra (bra‘((bra‘𝐵) ∘ (𝐶 ketbra 𝐷)))))
 
19.6.15  Positive operators (cont.)
 
Theoremleopg 29827* Ordering relation for positive operators. Definition of positive operator ordering in [Kreyszig] p. 470. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
((𝑇𝐴𝑈𝐵) → (𝑇op 𝑈 ↔ ((𝑈op 𝑇) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈op 𝑇)‘𝑥) ·ih 𝑥))))
 
Theoremleop 29828* Ordering relation for operators. Definition of positive operator ordering in [Kreyszig] p. 470. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇op 𝑈 ↔ ∀𝑥 ∈ ℋ 0 ≤ (((𝑈op 𝑇)‘𝑥) ·ih 𝑥)))
 
Theoremleop2 29829* Ordering relation for operators. Definition of operator ordering in [Young] p. 141. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇op 𝑈 ↔ ∀𝑥 ∈ ℋ ((𝑇𝑥) ·ih 𝑥) ≤ ((𝑈𝑥) ·ih 𝑥)))
 
Theoremleop3 29830 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.)
((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → (𝑇op 𝑈 ↔ 0hopop (𝑈op 𝑇)))
 
Theoremleoppos 29831* Binary relation defining a positive operator. Definition VI.1 of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → ( 0hopop 𝑇 ↔ ∀𝑥 ∈ ℋ 0 ≤ ((𝑇𝑥) ·ih 𝑥)))
 
Theoremleoprf2 29832 The ordering relation for operators is reflexive. (Contributed by NM, 24-Jul-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → 𝑇op 𝑇)
 
Theoremleoprf 29833 The ordering relation for operators is reflexive. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → 𝑇op 𝑇)
 
Theoremleopsq 29834 The square of a Hermitian operator is positive. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → 0hopop (𝑇𝑇))
 
Theorem0leop 29835 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.)
0hopop 0hop
 
Theoremidleop 29836 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.)
0hopop Iop
 
Theoremleopadd 29837 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.)
(((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ ( 0hopop 𝑇 ∧ 0hopop 𝑈)) → 0hopop (𝑇 +op 𝑈))
 
Theoremleopmuli 29838 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.)
(((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp) ∧ (0 ≤ 𝐴 ∧ 0hopop 𝑇)) → 0hopop (𝐴 ·op 𝑇))
 
Theoremleopmul 29839 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.)
((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 0 < 𝐴) → ( 0hopop 𝑇 ↔ 0hopop (𝐴 ·op 𝑇)))
 
Theoremleopmul2i 29840 Scalar product applied to operator ordering. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
(((𝐴 ∈ ℝ ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (0 ≤ 𝐴𝑇op 𝑈)) → (𝐴 ·op 𝑇) ≤op (𝐴 ·op 𝑈))
 
Theoremleoptri 29841 The positive operator ordering relation satisfies trichotomy. Exercise 1(iii) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
((𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) → ((𝑇op 𝑈𝑈op 𝑇) ↔ 𝑇 = 𝑈))
 
Theoremleoptr 29842 The positive operator ordering relation is transitive. Exercise 1(iv) of [Retherford] p. 49. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)
(((𝑆 ∈ HrmOp ∧ 𝑇 ∈ HrmOp ∧ 𝑈 ∈ HrmOp) ∧ (𝑆op 𝑇𝑇op 𝑈)) → 𝑆op 𝑈)
 
Theoremleopnmid 29843 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.)
((𝑇 ∈ HrmOp ∧ (normop𝑇) ∈ ℝ) → 𝑇op ((normop𝑇) ·op Iop ))
 
Theoremnmopleid 29844 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.)
((𝑇 ∈ HrmOp ∧ (normop𝑇) ∈ ℝ ∧ 𝑇 ≠ 0hop ) → ((1 / (normop𝑇)) ·op 𝑇) ≤op Iop )
 
Theoremopsqrlem1 29845* Lemma for opsqri . (Contributed by NM, 9-Aug-2006.) (New usage is discouraged.)
𝑇 ∈ HrmOp    &   (normop𝑇) ∈ ℝ    &    0hopop 𝑇    &   𝑅 = ((1 / (normop𝑇)) ·op 𝑇)    &   (𝑇 ≠ 0hop → ∃𝑢 ∈ HrmOp ( 0hopop 𝑢 ∧ (𝑢𝑢) = 𝑅))       (𝑇 ≠ 0hop → ∃𝑣 ∈ HrmOp ( 0hopop 𝑣 ∧ (𝑣𝑣) = 𝑇))
 
Theoremopsqrlem2 29846* Lemma for opsqri . 𝐹𝑁 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.)
𝑇 ∈ HrmOp    &   𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇op (𝑥𝑥)))))    &   𝐹 = seq1(𝑆, (ℕ × { 0hop }))       (𝐹‘1) = 0hop
 
Theoremopsqrlem3 29847* Lemma for opsqri . (Contributed by NM, 22-Aug-2006.) (New usage is discouraged.)
𝑇 ∈ HrmOp    &   𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇op (𝑥𝑥)))))    &   𝐹 = seq1(𝑆, (ℕ × { 0hop }))       ((𝐺 ∈ HrmOp ∧ 𝐻 ∈ HrmOp) → (𝐺𝑆𝐻) = (𝐺 +op ((1 / 2) ·op (𝑇op (𝐺𝐺)))))
 
Theoremopsqrlem4 29848* Lemma for opsqri . (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
𝑇 ∈ HrmOp    &   𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇op (𝑥𝑥)))))    &   𝐹 = seq1(𝑆, (ℕ × { 0hop }))       𝐹:ℕ⟶HrmOp
 
Theoremopsqrlem5 29849* Lemma for opsqri . (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
𝑇 ∈ HrmOp    &   𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇op (𝑥𝑥)))))    &   𝐹 = seq1(𝑆, (ℕ × { 0hop }))       (𝑁 ∈ ℕ → (𝐹‘(𝑁 + 1)) = ((𝐹𝑁) +op ((1 / 2) ·op (𝑇op ((𝐹𝑁) ∘ (𝐹𝑁))))))
 
Theoremopsqrlem6 29850* Lemma for opsqri . (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)
𝑇 ∈ HrmOp    &   𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇op (𝑥𝑥)))))    &   𝐹 = seq1(𝑆, (ℕ × { 0hop }))    &   𝑇op Iop       (𝑁 ∈ ℕ → (𝐹𝑁) ≤op Iop )
 
19.6.16  Projectors as operators
 
Theorempjhmopi 29851 A projector is a Hermitian operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
𝐻C       (proj𝐻) ∈ HrmOp
 
Theorempjlnopi 29852 A projector is a linear operator. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)
𝐻C       (proj𝐻) ∈ LinOp
 
Theorempjnmopi 29853 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.)
𝐻C       (𝐻 ≠ 0 → (normop‘(proj𝐻)) = 1)
 
Theorempjbdlni 29854 A projector is a bounded linear operator. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)
𝐻C       (proj𝐻) ∈ BndLinOp
 
Theorempjhmop 29855 A projection is a Hermitian operator. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
(𝐻C → (proj𝐻) ∈ HrmOp)
 
Theoremhmopidmchi 29856 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.)
𝑇 ∈ HrmOp    &   (𝑇𝑇) = 𝑇       ran 𝑇C
 
Theoremhmopidmpji 29857 An idempotent Hermitian operator is a projection operator. Theorem 26.4 of [Halmos] p. 44. (Halmos seems to omit the proof that 𝐻 is a closed subspace, which is not trivial as hmopidmchi 29856 shows.) (Contributed by NM, 22-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
𝑇 ∈ HrmOp    &   (𝑇𝑇) = 𝑇       𝑇 = (proj‘ran 𝑇)
 
Theoremhmopidmch 29858 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.)
((𝑇 ∈ HrmOp ∧ (𝑇𝑇) = 𝑇) → ran 𝑇C )
 
Theoremhmopidmpj 29859 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.)
((𝑇 ∈ HrmOp ∧ (𝑇𝑇) = 𝑇) → 𝑇 = (proj‘ran 𝑇))
 
Theorempjsdii 29860 Distributive law for Hilbert space operator sum. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)
𝐻C    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       ((proj𝐻) ∘ (𝑆 +op 𝑇)) = (((proj𝐻) ∘ 𝑆) +op ((proj𝐻) ∘ 𝑇))
 
Theorempjddii 29861 Distributive law for Hilbert space operator difference. (Contributed by NM, 24-Nov-2000.) (New usage is discouraged.)
𝐻C    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       ((proj𝐻) ∘ (𝑆op 𝑇)) = (((proj𝐻) ∘ 𝑆) −op ((proj𝐻) ∘ 𝑇))
 
Theorempjsdi2i 29862 Chained distributive law for Hilbert space operator difference. (Contributed by NM, 30-Nov-2000.) (New usage is discouraged.)
𝐻C    &   𝑅: ℋ⟶ ℋ    &   𝑆: ℋ⟶ ℋ    &   𝑇: ℋ⟶ ℋ       ((𝑅 ∘ (𝑆 +op 𝑇)) = ((𝑅𝑆) +op (𝑅𝑇)) → (((proj𝐻) ∘ 𝑅) ∘ (𝑆 +op 𝑇)) = ((((proj𝐻) ∘ 𝑅) ∘ 𝑆) +op (((proj𝐻) ∘ 𝑅) ∘ 𝑇)))
 
Theorempjcoi 29863 Composition of projections. (Contributed by NM, 16-Aug-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ ℋ → (((proj𝐺) ∘ (proj𝐻))‘𝐴) = ((proj𝐺)‘((proj𝐻)‘𝐴)))
 
Theorempjcocli 29864 Closure of composition of projections. (Contributed by NM, 29-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ ℋ → (((proj𝐺) ∘ (proj𝐻))‘𝐴) ∈ 𝐺)
 
Theorempjcohcli 29865 Closure of composition of projections. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ ℋ → (((proj𝐺) ∘ (proj𝐻))‘𝐴) ∈ ℋ)
 
Theorempjadjcoi 29866 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.)
𝐺C    &   𝐻C       ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((((proj𝐺) ∘ (proj𝐻))‘𝐴) ·ih 𝐵) = (𝐴 ·ih (((proj𝐻) ∘ (proj𝐺))‘𝐵)))
 
Theorempjcofni 29867 Functionality of composition of projections. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       ((proj𝐺) ∘ (proj𝐻)) Fn ℋ
 
Theorempjss1coi 29868 Subset relationship for projections. Theorem 4.5(i)<->(iii) of [Beran] p. 112. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺𝐻 ↔ ((proj𝐻) ∘ (proj𝐺)) = (proj𝐺))
 
Theorempjss2coi 29869 Subset relationship for projections. Theorem 4.5(i)<->(ii) of [Beran] p. 112. (Contributed by NM, 7-Oct-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺𝐻 ↔ ((proj𝐺) ∘ (proj𝐻)) = (proj𝐺))
 
Theorempjssmi 29870 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.)
𝐺C    &   𝐻C       (𝐴 ∈ ℋ → (𝐻𝐺 → (((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴)) = ((proj‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴)))
 
Theorempjssge0i 29871 Theorem 4.5(iv)->(v) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ ℋ → ((((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴)) = ((proj‘(𝐺 ∩ (⊥‘𝐻)))‘𝐴) → 0 ≤ ((((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴)) ·ih 𝐴)))
 
Theorempjdifnormi 29872 Theorem 4.5(v)<->(vi) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐴 ∈ ℋ → (0 ≤ ((((proj𝐺)‘𝐴) − ((proj𝐻)‘𝐴)) ·ih 𝐴) ↔ (norm‘((proj𝐻)‘𝐴)) ≤ (norm‘((proj𝐺)‘𝐴))))
 
Theorempjnormssi 29873* Theorem 4.5(i)<->(vi) of [Beran] p. 112. (Contributed by NM, 26-Sep-2001.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺𝐻 ↔ ∀𝑥 ∈ ℋ (norm‘((proj𝐺)‘𝑥)) ≤ (norm‘((proj𝐻)‘𝑥)))
 
Theorempjorthcoi 29874 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.)
𝐺C    &   𝐻C       (𝐺 ⊆ (⊥‘𝐻) → ((proj𝐺) ∘ (proj𝐻)) = 0hop )
 
Theorempjscji 29875 The projection of orthogonal subspaces is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺 ⊆ (⊥‘𝐻) → (proj‘(𝐺 𝐻)) = ((proj𝐺) +op (proj𝐻)))
 
Theorempjssumi 29876 The projection on a subspace sum is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺 ⊆ (⊥‘𝐻) → (proj‘(𝐺 + 𝐻)) = ((proj𝐺) +op (proj𝐻)))
 
Theorempjssposi 29877* 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.)
𝐺C    &   𝐻C       (∀𝑥 ∈ ℋ 0 ≤ ((((proj𝐻) −op (proj𝐺))‘𝑥) ·ih 𝑥) ↔ 𝐺𝐻)
 
Theorempjordi 29878* 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 𝐺𝐻; see pjssposi 29877). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
𝐺C    &   𝐻C       (∀𝑥 ∈ ℋ 0 ≤ ((((proj𝐻) −op (proj𝐺))‘𝑥) ·ih 𝑥) ↔ ((proj𝐺) “ ℋ) ⊆ ((proj𝐻) “ ℋ))
 
Theorempjssdif2i 29879 The projection subspace of the difference between two projectors. Part 2 of Theorem 29.3 of [Halmos] p. 48 (shortened with pjssposi 29877). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺𝐻 ↔ ((proj𝐻) −op (proj𝐺)) = (proj‘(𝐻 ∩ (⊥‘𝐺))))
 
Theorempjssdif1i 29880 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 29877). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺𝐻 ↔ ((proj𝐻) −op (proj𝐺)) ∈ ran proj)
 
Theorempjimai 29881 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.)
𝐴S    &   𝐵C       ((proj𝐵) “ 𝐴) = ((𝐴 + (⊥‘𝐵)) ∩ 𝐵)
 
Theorempjidmcoi 29882 A projection is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 1-Oct-2000.) (New usage is discouraged.)
𝐻C       ((proj𝐻) ∘ (proj𝐻)) = (proj𝐻)
 
Theorempjoccoi 29883 Composition of projections of a subspace and its orthocomplement. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)
𝐻C       ((proj𝐻) ∘ (proj‘(⊥‘𝐻))) = 0hop
 
Theorempjtoi 29884 Subspace sum of projection and projection of orthocomplement. (Contributed by NM, 16-Nov-2000.) (New usage is discouraged.)
𝐻C       ((proj𝐻) +op (proj‘(⊥‘𝐻))) = (proj‘ ℋ)
 
Theorempjoci 29885 Projection of orthocomplement. First part of Theorem 27.3 of [Halmos] p. 45. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)
𝐻C       ((proj‘ ℋ) −op (proj𝐻)) = (proj‘(⊥‘𝐻))
 
Theorempjidmco 29886 A projection operator is idempotent. Property (ii) of [Beran] p. 109. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
(𝐻C → ((proj𝐻) ∘ (proj𝐻)) = (proj𝐻))
 
Theoremdfpjop 29887 Definition of projection operator in [Hughes] p. 47, except that we do not need linearity to be explicit by virtue of hmoplin 29647. (Contributed by NM, 24-Apr-2006.) (Revised by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
(𝑇 ∈ ran proj ↔ (𝑇 ∈ HrmOp ∧ (𝑇𝑇) = 𝑇))
 
Theorempjhmopidm 29888 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.)
ran proj = {𝑡 ∈ HrmOp ∣ (𝑡𝑡) = 𝑡}
 
Theoremelpjidm 29889 A projection operator is idempotent. Part of Theorem 26.1 of [Halmos] p. 43. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
(𝑇 ∈ ran proj → (𝑇𝑇) = 𝑇)
 
Theoremelpjhmop 29890 A projection operator is Hermitian. Part of Theorem 26.1 of [Halmos] p. 43. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
(𝑇 ∈ ran proj𝑇 ∈ HrmOp)
 
Theorem0leopj 29891 A projector is a positive operator. (Contributed by NM, 27-Sep-2008.) (New usage is discouraged.)
(𝑇 ∈ ran proj → 0hopop 𝑇)
 
Theorempjadj2 29892 A projector is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 3-Jun-2006.) (New usage is discouraged.)
(𝑇 ∈ ran proj → (adj𝑇) = 𝑇)
 
Theorempjadj3 29893 A projector is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
(𝐻C → (adj‘(proj𝐻)) = (proj𝐻))
 
Theoremelpjch 29894 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.)
(𝑇 ∈ ran proj → (ran 𝑇C𝑇 = (proj‘ran 𝑇)))
 
Theoremelpjrn 29895* 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.)
(𝑇 ∈ ran proj → ran 𝑇 = {𝑥 ∈ ℋ ∣ (𝑇𝑥) = 𝑥})
 
Theorempjinvari 29896 A closed subspace 𝐻 with projection 𝑇 is invariant under an operator 𝑆 iff 𝑆𝑇 = 𝑇𝑆𝑇. Theorem 27.1 of [Halmos] p. 45. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)
𝑆: ℋ⟶ ℋ    &   𝐻C    &   𝑇 = (proj𝐻)       ((𝑆𝑇): ℋ⟶𝐻 ↔ (𝑆𝑇) = (𝑇 ∘ (𝑆𝑇)))
 
Theorempjin1i 29897 Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)
𝐺C    &   𝐻C       (proj‘(𝐺𝐻)) = ((proj𝐺) ∘ (proj‘(𝐺𝐻)))
 
Theorempjin2i 29898 Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)
𝐺C    &   𝐻C       (((proj𝐺) = ((proj𝐺) ∘ (proj𝐻)) ∧ (proj𝐻) = ((proj𝐻) ∘ (proj𝐺))) ↔ (proj𝐺) = (proj𝐻))
 
Theorempjin3i 29899 Lemma for Theorem 1.22 of Mittelstaedt, p. 20. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)
𝐹C    &   𝐺C    &   𝐻C       (((proj𝐹) = ((proj𝐹) ∘ (proj𝐺)) ∧ (proj𝐹) = ((proj𝐹) ∘ (proj𝐻))) ↔ (proj𝐹) = ((proj𝐹) ∘ (proj‘(𝐺𝐻))))
 
Theorempjclem1 29900 Lemma for projection commutation theorem. (Contributed by NM, 16-Nov-2000.) (New usage is discouraged.)
𝐺C    &   𝐻C       (𝐺 𝐶 𝐻 → ((proj𝐺) ∘ (proj𝐻)) = (proj‘(𝐺𝐻)))
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