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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | hopncani 29601 | Hilbert space operator cancellation law. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
⊢ 𝑇: ℋ⟶ ℋ & ⊢ 𝑈: ℋ⟶ ℋ ⇒ ⊢ ((𝑇 +op 𝑈) −op 𝑈) = 𝑇 | ||
Theorem | honpcani 29602 | Hilbert space operator cancellation law. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.) |
⊢ 𝑇: ℋ⟶ ℋ & ⊢ 𝑈: ℋ⟶ ℋ ⇒ ⊢ ((𝑇 −op 𝑈) +op 𝑈) = 𝑇 | ||
Theorem | hosubeq0i 29603 | If the difference between two operators is zero, they are equal. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.) |
⊢ 𝑇: ℋ⟶ ℋ & ⊢ 𝑈: ℋ⟶ ℋ ⇒ ⊢ ((𝑇 −op 𝑈) = 0hop ↔ 𝑇 = 𝑈) | ||
Theorem | honpncani 29604 | Hilbert space operator cancellation law. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.) |
⊢ 𝑅: ℋ⟶ ℋ & ⊢ 𝑆: ℋ⟶ ℋ & ⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ ((𝑅 −op 𝑆) +op (𝑆 −op 𝑇)) = (𝑅 −op 𝑇) | ||
Theorem | ho01i 29605* | A condition implying that a Hilbert space operator is identically zero. Lemma 3.2(S8) of [Beran] p. 95. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.) |
⊢ 𝑇: ℋ⟶ ℋ ⇒ ⊢ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = 0 ↔ 𝑇 = 0hop ) | ||
Theorem | ho02i 29606* | 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 (𝑇‘𝑦)) = 0 ↔ 𝑇 = 0hop ) | ||
Theorem | hoeq1 29607* | 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 ((𝑆‘𝑥) ·ih 𝑦) = ((𝑇‘𝑥) ·ih 𝑦) ↔ 𝑆 = 𝑇)) | ||
Theorem | hoeq2 29608* | 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 (𝑆‘𝑦)) = (𝑥 ·ih (𝑇‘𝑦)) ↔ 𝑆 = 𝑇)) | ||
Theorem | adjmo 29609* | 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 (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦)) | ||
Theorem | adjsym 29610* | 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 (𝑆‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑆‘𝑥) ·ih 𝑦))) | ||
Theorem | eigrei 29611 | 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 ⇒ ⊢ (((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ) → ((𝐴 ·ih (𝑇‘𝐴)) = ((𝑇‘𝐴) ·ih 𝐴) ↔ 𝐵 ∈ ℝ)) | ||
Theorem | eigre 29612 | 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) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → ((𝐴 ·ih (𝑇‘𝐴)) = ((𝑇‘𝐴) ·ih 𝐴) ↔ 𝐵 ∈ ℝ)) | ||
Theorem | eigposi 29613 | 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 (𝑇‘𝐴)) ∈ ℝ ∧ 0 ≤ (𝐴 ·ih (𝑇‘𝐴))) ∧ ((𝑇‘𝐴) = (𝐵 ·ℎ 𝐴) ∧ 𝐴 ≠ 0ℎ)) → (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) | ||
Theorem | eigorthi 29614 | 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 ⇒ ⊢ ((((𝑇‘𝐴) = (𝐶 ·ℎ 𝐴) ∧ (𝑇‘𝐵) = (𝐷 ·ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷)) → ((𝐴 ·ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵) ↔ (𝐴 ·ih 𝐵) = 0)) | ||
Theorem | eigorth 29615 | 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)) ∧ (((𝑇‘𝐴) = (𝐶 ·ℎ 𝐴) ∧ (𝑇‘𝐵) = (𝐷 ·ℎ 𝐵)) ∧ 𝐶 ≠ (∗‘𝐷))) → ((𝐴 ·ih (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵) ↔ (𝐴 ·ih 𝐵) = 0)) | ||
Definition | df-nmop 29616* | 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 ((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑡‘𝑧)))}, ℝ*, < )) | ||
Definition | df-cnop 29617* | 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 ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((𝑡‘𝑤) −ℎ (𝑡‘𝑥))) < 𝑦)} | ||
Definition | df-lnop 29618* | Define the set of linear operators on Hilbert space. (See df-hosum 29507 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 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑡‘𝑦)) +ℎ (𝑡‘𝑧))} | ||
Definition | df-bdop 29619 | Define the set of bounded linear Hilbert space operators. (See df-hosum 29507 for definition of operator.) (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.) |
⊢ BndLinOp = {𝑡 ∈ LinOp ∣ (normop‘𝑡) < +∞} | ||
Definition | df-unop 29620* | 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 ((𝑡‘𝑥) ·ih (𝑡‘𝑦)) = (𝑥 ·ih 𝑦))} | ||
Definition | df-hmop 29621* | 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 (𝑡‘𝑦)) = ((𝑡‘𝑥) ·ih 𝑦)} | ||
Definition | df-nmfn 29622* | 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 ((normℎ‘𝑧) ≤ 1 ∧ 𝑥 = (abs‘(𝑡‘𝑧)))}, ℝ*, < )) | ||
Definition | df-nlfn 29623 | 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})) | ||
Definition | df-cnfn 29624* | 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 ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (abs‘((𝑡‘𝑤) − (𝑡‘𝑥))) < 𝑦)} | ||
Definition | df-lnfn 29625* | 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 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑡‘𝑦)) + (𝑡‘𝑧))} | ||
Definition | df-adjh 29626* | Define the adjoint of a Hilbert space operator (if it exists). The domain of adjℎ 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 29860) that the adjoint exists for a bounded linear operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.) |
⊢ adjℎ = {〈𝑡, 𝑢〉 ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑡‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑢‘𝑦)))} | ||
Definition | df-bra 29627* |
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, 〈𝐴 ∣ 𝐵〉 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 ℋ to ℂ. We represent the
Dirac notation 〈𝐴 ∣ 𝐵〉 by ((bra‘𝐴)‘𝐵); see
braval 29721. The reversal of the inner product
arguments not only makes
the bra-ket behavior consistent with physics literature (see comments
under ax-his3 28861) but is also required in order for the
associative law
kbass2 29894 to work.
Our definition of bra and the associated outer product df-kb 29628 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 29628, under the 17-May-2006 entry. (Contributed by NM, 15-May-2006.) (New usage is discouraged.) |
⊢ bra = (𝑥 ∈ ℋ ↦ (𝑦 ∈ ℋ ↦ (𝑦 ·ih 𝑥))) | ||
Definition | df-kb 29628* | 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 29627, 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) ·ℎ 𝑥))) | ||
Definition | df-leop 29629* | Define positive operator ordering. Definition VI.1 of [Retherford] p. 49. Note that ( ℋ × 0ℋ) ≤op 𝑇 means that 𝑇 is a positive operator. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.) |
⊢ ≤op = {〈𝑡, 𝑢〉 ∣ ((𝑢 −op 𝑡) ∈ HrmOp ∧ ∀𝑥 ∈ ℋ 0 ≤ (((𝑢 −op 𝑡)‘𝑥) ·ih 𝑥))} | ||
Definition | df-eigvec 29630* | Define the eigenvector function. Theorem eleigveccl 29736 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 ∖ 0ℋ) ∣ ∃𝑧 ∈ ℂ (𝑡‘𝑥) = (𝑧 ·ℎ 𝑥)}) | ||
Definition | df-eigval 29631* | Define the eigenvalue function. The range of eigval‘𝑇 is the set of eigenvalues of Hilbert space operator 𝑇. Theorem eigvalcl 29738 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) ↦ (((𝑡‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2)))) | ||
Definition | df-spec 29632* | 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 ∣ ¬ (𝑡 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ}) | ||
Theorem | nmopval 29633* | 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 → (normop‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))}, ℝ*, < )) | ||
Theorem | elcnop 29634* | 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 ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (normℎ‘((𝑇‘𝑤) −ℎ (𝑇‘𝑥))) < 𝑦))) | ||
Theorem | ellnop 29635* | 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 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)))) | ||
Theorem | lnopf 29636 | 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) | ||
Theorem | elbdop 29637 | 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 ∧ (normop‘𝑇) < +∞)) | ||
Theorem | bdopln 29638 | A bounded linear Hilbert space operator is a linear operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp) | ||
Theorem | bdopf 29639 | 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) | ||
Theorem | nmopsetretALT 29640* | The set in the supremum of the operator norm definition df-nmop 29616 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 ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ) | ||
Theorem | nmopsetretHIL 29641* | The set in the supremum of the operator norm definition df-nmop 29616 is a set of reals. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.) |
⊢ (𝑇: ℋ⟶ ℋ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} ⊆ ℝ) | ||
Theorem | nmopsetn0 29642* | The set in the supremum of the operator norm definition df-nmop 29616 is nonempty. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.) |
⊢ (normℎ‘(𝑇‘0ℎ)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (normℎ‘(𝑇‘𝑦)))} | ||
Theorem | nmopxr 29643 | The norm of a Hilbert space operator is an extended real. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.) |
⊢ (𝑇: ℋ⟶ ℋ → (normop‘𝑇) ∈ ℝ*) | ||
Theorem | nmoprepnf 29644 | 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 → ((normop‘𝑇) ∈ ℝ ↔ (normop‘𝑇) ≠ +∞)) | ||
Theorem | nmopgtmnf 29645 | The norm of a Hilbert space operator is not minus infinity. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.) |
⊢ (𝑇: ℋ⟶ ℋ → -∞ < (normop‘𝑇)) | ||
Theorem | nmopreltpnf 29646 | 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 → ((normop‘𝑇) ∈ ℝ ↔ (normop‘𝑇) < +∞)) | ||
Theorem | nmopre 29647 | The norm of a bounded operator is a real number. (Contributed by NM, 29-Jan-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ BndLinOp → (normop‘𝑇) ∈ ℝ) | ||
Theorem | elbdop2 29648 | Property defining a bounded linear Hilbert space operator. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ BndLinOp ↔ (𝑇 ∈ LinOp ∧ (normop‘𝑇) ∈ ℝ)) | ||
Theorem | elunop 29649* | 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 ((𝑇‘𝑥) ·ih (𝑇‘𝑦)) = (𝑥 ·ih 𝑦))) | ||
Theorem | elhmop 29650* | 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 (𝑇‘𝑦)) = ((𝑇‘𝑥) ·ih 𝑦))) | ||
Theorem | hmopf 29651 | A Hermitian operator is a Hilbert space operator (mapping). (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ HrmOp → 𝑇: ℋ⟶ ℋ) | ||
Theorem | hmopex 29652 | The class of Hermitian operators is a set. (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.) |
⊢ HrmOp ∈ V | ||
Theorem | nmfnval 29653* | 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 → (normfn‘𝑇) = sup({𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))}, ℝ*, < )) | ||
Theorem | nmfnsetre 29654* | The set in the supremum of the functional norm definition df-nmfn 29622 is a set of reals. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
⊢ (𝑇: ℋ⟶ℂ → {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} ⊆ ℝ) | ||
Theorem | nmfnsetn0 29655* | The set in the supremum of the functional norm definition df-nmfn 29622 is nonempty. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
⊢ (abs‘(𝑇‘0ℎ)) ∈ {𝑥 ∣ ∃𝑦 ∈ ℋ ((normℎ‘𝑦) ≤ 1 ∧ 𝑥 = (abs‘(𝑇‘𝑦)))} | ||
Theorem | nmfnxr 29656 | The norm of any Hilbert space functional is an extended real. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.) |
⊢ (𝑇: ℋ⟶ℂ → (normfn‘𝑇) ∈ ℝ*) | ||
Theorem | nmfnrepnf 29657 | 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 → ((normfn‘𝑇) ∈ ℝ ↔ (normfn‘𝑇) ≠ +∞)) | ||
Theorem | nlfnval 29658 | 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})) | ||
Theorem | elcnfn 29659* | 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 ((normℎ‘(𝑤 −ℎ 𝑥)) < 𝑧 → (abs‘((𝑇‘𝑤) − (𝑇‘𝑥))) < 𝑦))) | ||
Theorem | ellnfn 29660* | 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 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (𝑇‘𝑦)) + (𝑇‘𝑧)))) | ||
Theorem | lnfnf 29661 | A linear Hilbert space functional is a functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ LinFn → 𝑇: ℋ⟶ℂ) | ||
Theorem | dfadj2 29662* | Alternate definition of the adjoint of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.) |
⊢ adjℎ = {〈𝑡, 𝑢〉 ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑡‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦))} | ||
Theorem | funadj 29663 | Functionality of the adjoint function. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
⊢ Fun adjℎ | ||
Theorem | dmadjss 29664 | The domain of the adjoint function is a subset of the maps from ℋ to ℋ. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
⊢ dom adjℎ ⊆ ( ℋ ↑m ℋ) | ||
Theorem | dmadjop 29665 | 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 adjℎ → 𝑇: ℋ⟶ ℋ) | ||
Theorem | adjeu 29666* | 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 adjℎ ↔ ∃!𝑢 ∈ ( ℋ ↑m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦))) | ||
Theorem | adjval 29667* | Value of the adjoint function for 𝑇 in the domain of adjℎ. (Contributed by NM, 19-Feb-2006.) (Revised by Mario Carneiro, 24-Dec-2016.) (New usage is discouraged.) |
⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇) = (℩𝑢 ∈ ( ℋ ↑m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦))) | ||
Theorem | adjval2 29668* | Value of the adjoint function. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇) = (℩𝑢 ∈ ( ℋ ↑m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑢‘𝑦)))) | ||
Theorem | cnvadj 29669 | The adjoint function equals its converse. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
⊢ ◡adjℎ = adjℎ | ||
Theorem | funcnvadj 29670 | The converse of the adjoint function is a function. (Contributed by NM, 25-Jan-2006.) (New usage is discouraged.) |
⊢ Fun ◡adjℎ | ||
Theorem | adj1o 29671 | The adjoint function maps one-to-one onto its domain. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
⊢ adjℎ:dom adjℎ–1-1-onto→dom adjℎ | ||
Theorem | dmadjrn 29672 | The adjoint of an operator belongs to the adjoint function's domain. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘𝑇) ∈ dom adjℎ) | ||
Theorem | eigvecval 29673* | 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 ∖ 0ℋ) ∣ ∃𝑦 ∈ ℂ (𝑇‘𝑥) = (𝑦 ·ℎ 𝑥)}) | ||
Theorem | eigvalfval 29674* | 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) ↦ (((𝑇‘𝑥) ·ih 𝑥) / ((normℎ‘𝑥)↑2)))) | ||
Theorem | specval 29675* | 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 ∣ ¬ (𝑇 −op (𝑥 ·op ( I ↾ ℋ))): ℋ–1-1→ ℋ}) | ||
Theorem | speccl 29676 | 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) | ||
Theorem | hhlnoi 29677 | The linear operators of Hilbert space. (Contributed by NM, 19-Nov-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝐿 = (𝑈 LnOp 𝑈) ⇒ ⊢ LinOp = 𝐿 | ||
Theorem | hhnmoi 29678 | 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 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑁 = (𝑈 normOpOLD 𝑈) ⇒ ⊢ normop = 𝑁 | ||
Theorem | hhbloi 29679 | 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 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝐵 = (𝑈 BLnOp 𝑈) ⇒ ⊢ BndLinOp = 𝐵 | ||
Theorem | hh0oi 29680 | The zero operator in Hilbert space. (Contributed by NM, 7-Dec-2007.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑍 = (𝑈 0op 𝑈) ⇒ ⊢ 0hop = 𝑍 | ||
Theorem | hhcno 29681 | The continuous operators of Hilbert space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) |
⊢ 𝐷 = (normℎ ∘ −ℎ ) & ⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ ContOp = (𝐽 Cn 𝐽) | ||
Theorem | hhcnf 29682 | The continuous functionals of Hilbert space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) |
⊢ 𝐷 = (normℎ ∘ −ℎ ) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝐾 = (TopOpen‘ℂfld) ⇒ ⊢ ContFn = (𝐽 Cn 𝐾) | ||
Theorem | dmadjrnb 29683 | 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 6700.) (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ dom adjℎ ↔ (adjℎ‘𝑇) ∈ dom adjℎ) | ||
Theorem | nmoplb 29684 | A lower bound for an operator norm. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (normℎ‘(𝑇‘𝐴)) ≤ (normop‘𝑇)) | ||
Theorem | nmopub 29685* | An upper bound for an operator norm. (Contributed by NM, 7-Mar-2006.) (New usage is discouraged.) |
⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝐴 ∈ ℝ*) → ((normop‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (normℎ‘(𝑇‘𝑥)) ≤ 𝐴))) | ||
Theorem | nmopub2tALT 29686* | 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 (normℎ‘(𝑇‘𝑥)) ≤ (𝐴 · (normℎ‘𝑥))) → (normop‘𝑇) ≤ 𝐴) | ||
Theorem | nmopub2tHIL 29687* | 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 (normℎ‘(𝑇‘𝑥)) ≤ (𝐴 · (normℎ‘𝑥))) → (normop‘𝑇) ≤ 𝐴) | ||
Theorem | nmopge0 29688 | The norm of any Hilbert space operator is nonnegative. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.) |
⊢ (𝑇: ℋ⟶ ℋ → 0 ≤ (normop‘𝑇)) | ||
Theorem | nmopgt0 29689 | 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 → ((normop‘𝑇) ≠ 0 ↔ 0 < (normop‘𝑇))) | ||
Theorem | cnopc 29690* | 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 ((normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥 → (normℎ‘((𝑇‘𝑦) −ℎ (𝑇‘𝐴))) < 𝐵)) | ||
Theorem | lnopl 29691 | 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 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘𝐶))) | ||
Theorem | unop 29692 | Basic inner product property of a unitary operator. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.) |
⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) ·ih (𝑇‘𝐵)) = (𝐴 ·ih 𝐵)) | ||
Theorem | unopf1o 29693 | 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) | ||
Theorem | unopnorm 29694 | A unitary operator is idempotent in the norm. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.) |
⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ) → (normℎ‘(𝑇‘𝐴)) = (normℎ‘𝐴)) | ||
Theorem | cnvunop 29695 | 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) | ||
Theorem | unopadj 29696 | 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) → ((𝑇‘𝐴) ·ih 𝐵) = (𝐴 ·ih (◡𝑇‘𝐵))) | ||
Theorem | unoplin 29697 | A unitary operator is linear. Theorem in [AkhiezerGlazman] p. 72. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.) |
⊢ (𝑇 ∈ UniOp → 𝑇 ∈ LinOp) | ||
Theorem | counop 29698 | The composition of two unitary operators is unitary. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.) |
⊢ ((𝑆 ∈ UniOp ∧ 𝑇 ∈ UniOp) → (𝑆 ∘ 𝑇) ∈ UniOp) | ||
Theorem | hmop 29699 | 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 (𝑇‘𝐵)) = ((𝑇‘𝐴) ·ih 𝐵)) | ||
Theorem | hmopre 29700 | 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) → ((𝑇‘𝐴) ·ih 𝐴) ∈ ℝ) |
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