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Theorem List for Metamath Proof Explorer - 29701-29800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremho02i 29701* A condition implying that a Hilbert space operator is identically zero. Lemma 3.2(S10) of [Beran] p. 95. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
𝑇: ℋ⟶ ℋ       (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = 0 ↔ 𝑇 = 0hop )

Theoremhoeq1 29702* A condition implying that two Hilbert space operators are equal. Lemma 3.2(S9) of [Beran] p. 95. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑆𝑥) ·ih 𝑦) = ((𝑇𝑥) ·ih 𝑦) ↔ 𝑆 = 𝑇))

Theoremhoeq2 29703* A condition implying that two Hilbert space operators are equal. Lemma 3.2(S11) of [Beran] p. 95. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = (𝑥 ·ih (𝑇𝑦)) ↔ 𝑆 = 𝑇))

Theoremadjmo 29704* Every Hilbert space operator has at most one adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
∃*𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦))

Theoremadjsym 29705* Symmetry property of an adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑆𝑦)) = ((𝑇𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑆𝑥) ·ih 𝑦)))

Theoremeigrei 29706 A necessary and sufficient condition (that holds when 𝑇 is a Hermitian operator) for an eigenvalue 𝐵 to be real. Generalization of Equation 1.30 of [Hughes] p. 49. (Contributed by NM, 21-Jan-2005.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℂ       (((𝑇𝐴) = (𝐵 · 𝐴) ∧ 𝐴 ≠ 0) → ((𝐴 ·ih (𝑇𝐴)) = ((𝑇𝐴) ·ih 𝐴) ↔ 𝐵 ∈ ℝ))

Theoremeigre 29707 A necessary and sufficient condition (that holds when 𝑇 is a Hermitian operator) for an eigenvalue 𝐵 to be real. Generalization of Equation 1.30 of [Hughes] p. 49. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)
(((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℂ) ∧ ((𝑇𝐴) = (𝐵 · 𝐴) ∧ 𝐴 ≠ 0)) → ((𝐴 ·ih (𝑇𝐴)) = ((𝑇𝐴) ·ih 𝐴) ↔ 𝐵 ∈ ℝ))

Theoremeigposi 29708 A sufficient condition (first conjunct pair, that holds when 𝑇 is a positive operator) for an eigenvalue 𝐵 (second conjunct pair) to be nonnegative. Remark (ii) in [Hughes] p. 137. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℂ       ((((𝐴 ·ih (𝑇𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇𝐴))) ∧ ((𝑇𝐴) = (𝐵 · 𝐴) ∧ 𝐴 ≠ 0)) → (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵))

Theoremeigorthi 29709 A necessary and sufficient condition (that holds when 𝑇 is a Hermitian operator) for two eigenvectors 𝐴 and 𝐵 to be orthogonal. Generalization of Equation 1.31 of [Hughes] p. 49. (Contributed by NM, 23-Jan-2005.) (New usage is discouraged.)
𝐴 ∈ ℋ    &   𝐵 ∈ ℋ    &   𝐶 ∈ ℂ    &   𝐷 ∈ ℂ       ((((𝑇𝐴) = (𝐶 · 𝐴) ∧ (𝑇𝐵) = (𝐷 · 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) → ((𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵) ↔ (𝐴 ·ih 𝐵) = 0))

Theoremeigorth 29710 A necessary and sufficient condition (that holds when 𝑇 is a Hermitian operator) for two eigenvectors 𝐴 and 𝐵 to be orthogonal. Generalization of Equation 1.31 of [Hughes] p. 49. (Contributed by NM, 23-Mar-2006.) (New usage is discouraged.)
((((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) ∧ (((𝑇𝐴) = (𝐶 · 𝐴) ∧ (𝑇𝐵) = (𝐷 · 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷))) → ((𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵) ↔ (𝐴 ·ih 𝐵) = 0))

19.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms

Definitiondf-nmop 29711* Define the norm of a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
normop = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((norm𝑧) ≤ 1 ∧ 𝑥 = (norm‘(𝑡𝑧)))}, ℝ*, < ))

Definitiondf-cnop 29712* Define the set of continuous operators on Hilbert space. For every "epsilon" (𝑦) there is a "delta" (𝑧) such that... (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
ContOp = {𝑡 ∈ ( ℋ ↑m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((𝑡𝑤) − (𝑡𝑥))) < 𝑦)}

Definitiondf-lnop 29713* Define the set of linear operators on Hilbert space. (See df-hosum 29602 for definition of operator.) (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
LinOp = {𝑡 ∈ ( ℋ ↑m ℋ) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑡𝑦)) + (𝑡𝑧))}

Definitiondf-bdop 29714 Define the set of bounded linear Hilbert space operators. (See df-hosum 29602 for definition of operator.) (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
BndLinOp = {𝑡 ∈ LinOp ∣ (normop𝑡) < +∞}

Definitiondf-unop 29715* Define the set of unitary operators on Hilbert space. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
UniOp = {𝑡 ∣ (𝑡: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡𝑥) ·ih (𝑡𝑦)) = (𝑥 ·ih 𝑦))}

Definitiondf-hmop 29716* Define the set of Hermitian operators on Hilbert space. Some books call these "symmetric operators" and others call them "self-adjoint operators", sometimes with slightly different technical meanings. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
HrmOp = {𝑡 ∈ ( ℋ ↑m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡𝑦)) = ((𝑡𝑥) ·ih 𝑦)}

19.6.5  Linear and continuous functionals and norms

Definitiondf-nmfn 29717* Define the norm of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
normfn = (𝑡 ∈ (ℂ ↑m ℋ) ↦ sup({𝑥 ∣ ∃𝑧 ∈ ℋ ((norm𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡𝑧)))}, ℝ*, < ))

Definitiondf-nlfn 29718 Define the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
null = (𝑡 ∈ (ℂ ↑m ℋ) ↦ (𝑡 “ {0}))

Definitiondf-cnfn 29719* Define the set of continuous functionals on Hilbert space. For every "epsilon" (𝑦) there is a "delta" (𝑧) such that... (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
ContFn = {𝑡 ∈ (ℂ ↑m ℋ) ∣ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (abs‘((𝑡𝑤) − (𝑡𝑥))) < 𝑦)}

Definitiondf-lnfn 29720* Define the set of linear functionals on Hilbert space. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
LinFn = {𝑡 ∈ (ℂ ↑m ℋ) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑡𝑦)) + (𝑡𝑧))}

Definitiondf-adjh 29721* Define the adjoint of a Hilbert space operator (if it exists). The domain of adj is the set of all adjoint operators. Definition of adjoint in [Kalmbach2] p. 8. Unlike Kalmbach (and most authors), we do not demand that the operator be linear, but instead show (in adjbdln 29955) that the adjoint exists for a bounded linear operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
adj = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑢𝑦)))}

19.6.7  Dirac bra-ket notation

Definitiondf-bra 29722* Define the bra of a vector used by Dirac notation. Based on definition of bra in [Prugovecki] p. 186 (p. 180 in 1971 edition). In Dirac bra-ket notation, 𝐴𝐵 is a complex number equal to the inner product (𝐵 ·ih 𝐴). But physicists like to talk about the individual components 𝐴 and 𝐵, called bra and ket respectively. In order for their properties to make sense formally, we define the ket 𝐵 as the vector 𝐵 itself, and the bra 𝐴 as a functional from to . We represent the Dirac notation 𝐴𝐵 by ((bra‘𝐴)‘𝐵); see braval 29816. The reversal of the inner product arguments not only makes the bra-ket behavior consistent with physics literature (see comments under ax-his3 28956) but is also required in order for the associative law kbass2 29989 to work.

Our definition of bra and the associated outer product df-kb 29723 differs from, but is equivalent to, a common approach in the literature that makes use of mappings to a dual space. Our approach eliminates the need to have a parallel development of this dual space and instead keeps everything in Hilbert space.

For an extensive discussion about how our notation maps to the bra-ket notation in physics textbooks, see mmnotes.txt 29723, under the 17-May-2006 entry. (Contributed by NM, 15-May-2006.) (New usage is discouraged.)

bra = (𝑥 ∈ ℋ ↦ (𝑦 ∈ ℋ ↦ (𝑦 ·ih 𝑥)))

Definitiondf-kb 29723* Define a commuted bra and ket juxtaposition used by Dirac notation. In Dirac notation, 𝐴⟩⟨𝐵 is an operator known as the outer product of 𝐴 and 𝐵, which we represent by (𝐴 ketbra 𝐵). Based on Equation 8.1 of [Prugovecki] p. 376. This definition, combined with Definition df-bra 29722, allows any legal juxtaposition of bras and kets to make sense formally and also to obey the associative law when mapped back to Dirac notation. (Contributed by NM, 15-May-2006.) (New usage is discouraged.)
ketbra = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) · 𝑥)))

19.6.8  Positive operators

Definitiondf-leop 29724* Define positive operator ordering. Definition VI.1 of [Retherford] p. 49. Note that ( ℋ × 0) ≤op 𝑇 means that 𝑇 is a positive operator. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
op = {⟨𝑡, 𝑢⟩ ∣ ((𝑢op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢op 𝑡)‘𝑥) ·ih 𝑥))}

19.6.9  Eigenvectors, eigenvalues, spectrum

Definitiondf-eigvec 29725* Define the eigenvector function. Theorem eleigveccl 29831 shows that eigvec‘𝑇, the set of eigenvectors of Hilbert space operator 𝑇, are Hilbert space vectors. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
eigvec = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ {𝑥 ∈ ( ℋ ∖ 0) ∣ ∃𝑧 ∈ ℂ (𝑡𝑥) = (𝑧 · 𝑥)})

Definitiondf-eigval 29726* Define the eigenvalue function. The range of eigval‘𝑇 is the set of eigenvalues of Hilbert space operator 𝑇. Theorem eigvalcl 29833 shows that (eigval‘𝑇)‘𝐴, the eigenvalue associated with eigenvector 𝐴, is a complex number. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
eigval = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ (𝑥 ∈ (eigvec‘𝑡) ↦ (((𝑡𝑥) ·ih 𝑥) / ((norm𝑥)↑2))))

Definitiondf-spec 29727* Define the spectrum of an operator. Definition of spectrum in [Halmos] p. 50. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
Lambda = (𝑡 ∈ ( ℋ ↑m ℋ) ↦ {𝑥 ∈ ℂ ∣ ¬ (𝑡op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ})

19.6.10  Theorems about operators and functionals

Theoremnmopval 29728* Value of the norm of a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (normop𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}, ℝ*, < ))

Theoremelcnop 29729* Property defining a continuous Hilbert space operator. (Contributed by NM, 28-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝑇 ∈ ContOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (norm‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦)))

Theoremellnop 29730* Property defining a linear Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝑇 ∈ LinOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))

Theoremlnopf 29731 A linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
(𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ)

Theoremelbdop 29732 Property defining a bounded linear Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop𝑇) < +∞))

Theorembdopln 29733 A bounded linear Hilbert space operator is a linear operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp)

Theorembdopf 29734 A bounded linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ)

TheoremnmopsetretALT 29735* The set in the supremum of the operator norm definition df-nmop 29711 is a set of reals. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))} ⊆ ℝ)

TheoremnmopsetretHIL 29736* The set in the supremum of the operator norm definition df-nmop 29711 is a set of reals. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))} ⊆ ℝ)

Theoremnmopsetn0 29737* The set in the supremum of the operator norm definition df-nmop 29711 is nonempty. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)
(norm‘(𝑇‘0)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (norm‘(𝑇𝑦)))}

Theoremnmopxr 29738 The norm of a Hilbert space operator is an extended real. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (normop𝑇) ∈ ℝ*)

Theoremnmoprepnf 29739 The norm of a Hilbert space operator is either real or plus infinity. (Contributed by NM, 5-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → ((normop𝑇) ∈ ℝ ↔ (normop𝑇) ≠ +∞))

Theoremnmopgtmnf 29740 The norm of a Hilbert space operator is not minus infinity. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → -∞ < (normop𝑇))

Theoremnmopreltpnf 29741 The norm of a Hilbert space operator is real iff it is less than infinity. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → ((normop𝑇) ∈ ℝ ↔ (normop𝑇) < +∞))

Theoremnmopre 29742 The norm of a bounded operator is a real number. (Contributed by NM, 29-Jan-2006.) (New usage is discouraged.)
(𝑇 ∈ BndLinOp → (normop𝑇) ∈ ℝ)

Theoremelbdop2 29743 Property defining a bounded linear Hilbert space operator. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop𝑇) ∈ ℝ))

Theoremelunop 29744* Property defining a unitary Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)
(𝑇 ∈ UniOp ↔ (𝑇: ℋ–onto→ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih (𝑇𝑦)) = (𝑥 ·ih 𝑦)))

Theoremelhmop 29745* Property defining a Hermitian Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝑇 ∈ HrmOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑇𝑥) ·ih 𝑦)))

Theoremhmopf 29746 A Hermitian operator is a Hilbert space operator (mapping). (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)
(𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ)

Theoremhmopex 29747 The class of Hermitian operators is a set. (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)
HrmOp ∈ V

Theoremnmfnval 29748* Value of the norm of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝑇: ℋ⟶ℂ → (normfn𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))}, ℝ*, < ))

Theoremnmfnsetre 29749* The set in the supremum of the functional norm definition df-nmfn 29717 is a set of reals. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ℂ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))} ⊆ ℝ)

Theoremnmfnsetn0 29750* The set in the supremum of the functional norm definition df-nmfn 29717 is nonempty. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
(abs‘(𝑇‘0)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((norm𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇𝑦)))}

Theoremnmfnxr 29751 The norm of any Hilbert space functional is an extended real. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ℂ → (normfn𝑇) ∈ ℝ*)

Theoremnmfnrepnf 29752 The norm of a Hilbert space functional is either real or plus infinity. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)
(𝑇: ℋ⟶ℂ → ((normfn𝑇) ∈ ℝ ↔ (normfn𝑇) ≠ +∞))

Theoremnlfnval 29753 Value of the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ℂ → (null‘𝑇) = (𝑇 “ {0}))

Theoremelcnfn 29754* Property defining a continuous functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝑇 ∈ ContFn ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℋ ((norm‘(𝑤 𝑥)) < 𝑧 → (abs‘((𝑇𝑤) − (𝑇𝑥))) < 𝑦)))

Theoremellnfn 29755* Property defining a linear functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝑇 ∈ LinFn ↔ (𝑇: ℋ⟶ℂ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 · (𝑇𝑦)) + (𝑇𝑧))))

Theoremlnfnf 29756 A linear Hilbert space functional is a functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
(𝑇 ∈ LinFn → 𝑇: ℋ⟶ℂ)

Theoremdfadj2 29757* Alternate definition of the adjoint of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)
adj = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡𝑦)) = ((𝑢𝑥) ·ih 𝑦))}

Theoremfunadj 29758 Functionality of the adjoint function. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Theoremdmadjss 29759 The domain of the adjoint function is a subset of the maps from to . (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
dom adj ⊆ ( ℋ ↑m ℋ)

Theoremdmadjop 29760 A member of the domain of the adjoint function is a Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ dom adj𝑇: ℋ⟶ ℋ)

Theoremadjeu 29761* Elementhood in the domain of the adjoint function. (Contributed by Mario Carneiro, 11-Sep-2015.) (Revised by Mario Carneiro, 24-Dec-2016.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (𝑇 ∈ dom adj ↔ ∃!𝑢 ∈ ( ℋ ↑m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)))

Theoremadjval 29762* Value of the adjoint function for 𝑇 in the domain of adj. (Contributed by NM, 19-Feb-2006.) (Revised by Mario Carneiro, 24-Dec-2016.) (New usage is discouraged.)
(𝑇 ∈ dom adj → (adj𝑇) = (𝑢 ∈ ( ℋ ↑m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇𝑦)) = ((𝑢𝑥) ·ih 𝑦)))

Theoremadjval2 29763* Value of the adjoint function. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)
(𝑇 ∈ dom adj → (adj𝑇) = (𝑢 ∈ ( ℋ ↑m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑢𝑦))))

Theoremcnvadj 29764 The adjoint function equals its converse. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Theoremfuncnvadj 29765 The converse of the adjoint function is a function. (Contributed by NM, 25-Jan-2006.) (New usage is discouraged.)

Theoremadj1o 29766 The adjoint function maps one-to-one onto its domain. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Theoremdmadjrn 29767 The adjoint of an operator belongs to the adjoint function's domain. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Theoremeigvecval 29768* The set of eigenvectors of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (eigvec‘𝑇) = {𝑥 ∈ ( ℋ ∖ 0) ∣ ∃𝑦 ∈ ℂ (𝑇𝑥) = (𝑦 · 𝑥)})

Theoremeigvalfval 29769* The eigenvalues of eigenvectors of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (eigval‘𝑇) = (𝑥 ∈ (eigvec‘𝑇) ↦ (((𝑇𝑥) ·ih 𝑥) / ((norm𝑥)↑2))))

Theoremspecval 29770* The value of the spectrum of an operator. (Contributed by NM, 11-Apr-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) = {𝑥 ∈ ℂ ∣ ¬ (𝑇op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ})

Theoremspeccl 29771 The spectrum of an operator is a set of complex numbers. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → (Lambda‘𝑇) ⊆ ℂ)

Theoremhhlnoi 29772 The linear operators of Hilbert space. (Contributed by NM, 19-Nov-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝐿 = (𝑈 LnOp 𝑈)       LinOp = 𝐿

Theoremhhnmoi 29773 The norm of an operator in Hilbert space. (Contributed by NM, 19-Nov-2007.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑁 = (𝑈 normOpOLD 𝑈)       normop = 𝑁

Theoremhhbloi 29774 A bounded linear operator in Hilbert space. (Contributed by NM, 19-Nov-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝐵 = (𝑈 BLnOp 𝑈)       BndLinOp = 𝐵

Theoremhh0oi 29775 The zero operator in Hilbert space. (Contributed by NM, 7-Dec-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, norm    &   𝑍 = (𝑈 0op 𝑈)        0hop = 𝑍

Theoremhhcno 29776 The continuous operators of Hilbert space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
𝐷 = (norm ∘ − )    &   𝐽 = (MetOpen‘𝐷)       ContOp = (𝐽 Cn 𝐽)

Theoremhhcnf 29777 The continuous functionals of Hilbert space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
𝐷 = (norm ∘ − )    &   𝐽 = (MetOpen‘𝐷)    &   𝐾 = (TopOpen‘ℂfld)       ContFn = (𝐽 Cn 𝐾)

Theoremdmadjrnb 29778 The adjoint of an operator belongs to the adjoint function's domain. (Note: the converse is dependent on our definition of function value, since it uses ndmfv 6686.) (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)

Theoremnmoplb 29779 A lower bound for an operator norm. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → (norm‘(𝑇𝐴)) ≤ (normop𝑇))

Theoremnmopub 29780* An upper bound for an operator norm. (Contributed by NM, 7-Mar-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → ((normop𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (norm‘(𝑇𝑥)) ≤ 𝐴)))

Theoremnmopub2tALT 29781* An upper bound for an operator norm. (Contributed by NM, 12-Apr-2006.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) ≤ (𝐴 · (norm𝑥))) → (normop𝑇) ≤ 𝐴)

Theoremnmopub2tHIL 29782* An upper bound for an operator norm. (Contributed by NM, 13-Dec-2007.) (New usage is discouraged.)
((𝑇: ℋ⟶ ℋ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥 ∈ ℋ (norm‘(𝑇𝑥)) ≤ (𝐴 · (norm𝑥))) → (normop𝑇) ≤ 𝐴)

Theoremnmopge0 29783 The norm of any Hilbert space operator is nonnegative. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → 0 ≤ (normop𝑇))

Theoremnmopgt0 29784 A linear Hilbert space operator that is not identically zero has a positive norm. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ ℋ → ((normop𝑇) ≠ 0 ↔ 0 < (normop𝑇)))

Theoremcnopc 29785* Basic continuity property of a continuous Hilbert space operator. (Contributed by NM, 2-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
((𝑇 ∈ ContOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℝ+𝑦 ∈ ℋ ((norm‘(𝑦 𝐴)) < 𝑥 → (norm‘((𝑇𝑦) − (𝑇𝐴))) < 𝐵))

Theoremlnopl 29786 Basic linearity property of a linear Hilbert space operator. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
(((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ) ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝑇‘((𝐴 · 𝐵) + 𝐶)) = ((𝐴 · (𝑇𝐵)) + (𝑇𝐶)))

Theoremunop 29787 Basic inner product property of a unitary operator. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇𝐴) ·ih (𝑇𝐵)) = (𝐴 ·ih 𝐵))

Theoremunopf1o 29788 A unitary operator in Hilbert space is one-to-one and onto. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
(𝑇 ∈ UniOp → 𝑇: ℋ–1-1-onto→ ℋ)

Theoremunopnorm 29789 A unitary operator is idempotent in the norm. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ) → (norm‘(𝑇𝐴)) = (norm𝐴))

Theoremcnvunop 29790 The inverse (converse) of a unitary operator in Hilbert space is unitary. Theorem in [AkhiezerGlazman] p. 72. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
(𝑇 ∈ UniOp → 𝑇 ∈ UniOp)

Theoremunopadj 29791 The inverse (converse) of a unitary operator is its adjoint. Equation 2 of [AkhiezerGlazman] p. 72. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇𝐴) ·ih 𝐵) = (𝐴 ·ih (𝑇𝐵)))

Theoremunoplin 29792 A unitary operator is linear. Theorem in [AkhiezerGlazman] p. 72. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
(𝑇 ∈ UniOp → 𝑇 ∈ LinOp)

Theoremcounop 29793 The composition of two unitary operators is unitary. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)
((𝑆 ∈ UniOp ∧ 𝑇 ∈ UniOp) → (𝑆𝑇) ∈ UniOp)

Theoremhmop 29794 Basic inner product property of a Hermitian operator. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)
((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇𝐵)) = ((𝑇𝐴) ·ih 𝐵))

Theoremhmopre 29795 The inner product of the value and argument of a Hermitian operator is real. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)
((𝑇 ∈ HrmOp ∧ 𝐴 ∈ ℋ) → ((𝑇𝐴) ·ih 𝐴) ∈ ℝ)

Theoremnmfnlb 29796 A lower bound for a functional norm. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (norm𝐴) ≤ 1) → (abs‘(𝑇𝐴)) ≤ (normfn𝑇))

Theoremnmfnleub 29797* An upper bound for the norm of a functional. (Contributed by NM, 24-May-2006.) (Revised by Mario Carneiro, 7-Sep-2014.) (New usage is discouraged.)
((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℝ*) → ((normfn𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((norm𝑥) ≤ 1 → (abs‘(𝑇𝑥)) ≤ 𝐴)))

Theoremnmfnleub2 29798* An upper bound for the norm of a functional. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)
((𝑇: ℋ⟶ℂ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ ∀𝑥 ∈ ℋ (abs‘(𝑇𝑥)) ≤ (𝐴 · (norm𝑥))) → (normfn𝑇) ≤ 𝐴)

Theoremnmfnge0 29799 The norm of any Hilbert space functional is nonnegative. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)
(𝑇: ℋ⟶ℂ → 0 ≤ (normfn𝑇))

Theoremelnlfn 29800 Membership in the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
(𝑇: ℋ⟶ℂ → (𝐴 ∈ (null‘𝑇) ↔ (𝐴 ∈ ℋ ∧ (𝑇𝐴) = 0)))

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