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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Definition | df-unop 31901* | 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 31902* | 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 31903* | 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 31904 | 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 31905* | 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 31906* | 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 31907* | 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 32141) 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 31908* |
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 32002. The reversal of the inner product
arguments not only makes
the bra-ket behavior consistent with physics literature (see comments
under ax-his3 31142) but is also required in order for the
associative law
kbass2 32175 to work.
Our definition of bra and the associated outer product df-kb 31909 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 31909, 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 31909* | 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 31908, 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 31910* | 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 31911* | Define the eigenvector function. Theorem eleigveccl 32017 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 31912* | Define the eigenvalue function. The range of eigval‘𝑇 is the set of eigenvalues of Hilbert space operator 𝑇. Theorem eigvalcl 32019 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 31913* | 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 31914* | 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 31915* | 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 31916* | 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 31917 | 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 31918 | 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 31919 | 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 31920 | 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 31921* | The set in the supremum of the operator norm definition df-nmop 31897 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 31922* | The set in the supremum of the operator norm definition df-nmop 31897 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 31923* | The set in the supremum of the operator norm definition df-nmop 31897 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 31924 | 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 31925 | 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 31926 | 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 31927 | 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 31928 | 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 31929 | 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 31930* | 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 31931* | 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 31932 | 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 31933 | The class of Hermitian operators is a set. (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.) |
| ⊢ HrmOp ∈ V | ||
| Theorem | nmfnval 31934* | 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 31935* | The set in the supremum of the functional norm definition df-nmfn 31903 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 31936* | The set in the supremum of the functional norm definition df-nmfn 31903 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 31937 | 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 31938 | 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 31939 | 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 31940* | 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 31941* | 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 31942 | 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 31943* | 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 31944 | Functionality of the adjoint function. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
| ⊢ Fun adjℎ | ||
| Theorem | dmadjss 31945 | 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 31946 | 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 31947* | 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 31948* | 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 31949* | 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 31950 | The adjoint function equals its converse. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
| ⊢ ◡adjℎ = adjℎ | ||
| Theorem | funcnvadj 31951 | The converse of the adjoint function is a function. (Contributed by NM, 25-Jan-2006.) (New usage is discouraged.) |
| ⊢ Fun ◡adjℎ | ||
| Theorem | adj1o 31952 | 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 31953 | 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 31954* | 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 31955* | 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 31956* | 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 31957 | 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 31958 | 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 31959 | 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 31960 | 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 31961 | 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 31962 | 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 31963 | 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 31964 | 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 6867.) (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ dom adjℎ ↔ (adjℎ‘𝑇) ∈ dom adjℎ) | ||
| Theorem | nmoplb 31965 | 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 31966* | 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 31967* | 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 31968* | 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 31969 | 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 31970 | 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 31971* | 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 31972 | 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 31973 | 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 31974 | 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 31975 | 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 31976 | 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 31977 | 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 31978 | 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 31979 | 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 31980 | 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 31981 | 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) | ||
| Theorem | nmfnlb 31982 | A lower bound for a functional norm. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (abs‘(𝑇‘𝐴)) ≤ (normfn‘𝑇)) | ||
| Theorem | nmfnleub 31983* | 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*) → ((normfn‘𝑇) ≤ 𝐴 ↔ ∀𝑥 ∈ ℋ ((normℎ‘𝑥) ≤ 1 → (abs‘(𝑇‘𝑥)) ≤ 𝐴))) | ||
| Theorem | nmfnleub2 31984* | 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‘(𝑇‘𝑥)) ≤ (𝐴 · (normℎ‘𝑥))) → (normfn‘𝑇) ≤ 𝐴) | ||
| Theorem | nmfnge0 31985 | The norm of any Hilbert space functional is nonnegative. (Contributed by NM, 24-May-2006.) (New usage is discouraged.) |
| ⊢ (𝑇: ℋ⟶ℂ → 0 ≤ (normfn‘𝑇)) | ||
| Theorem | elnlfn 31986 | 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 ∧ (𝑇‘𝐴) = 0))) | ||
| Theorem | elnlfn2 31987 | 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)) → (𝑇‘𝐴) = 0) | ||
| Theorem | cnfnc 31988* | Basic continuity property of a continuous functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ ContFn ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℝ+) → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ ℋ ((normℎ‘(𝑦 −ℎ 𝐴)) < 𝑥 → (abs‘((𝑇‘𝑦) − (𝑇‘𝐴))) < 𝐵)) | ||
| Theorem | lnfnl 31989 | Basic linearity property of a linear functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.) |
| ⊢ (((𝑇 ∈ LinFn ∧ 𝐴 ∈ ℂ) ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 𝐶)) = ((𝐴 · (𝑇‘𝐵)) + (𝑇‘𝐶))) | ||
| Theorem | adjcl 31990 | Closure of the adjoint of a Hilbert space operator. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ) → ((adjℎ‘𝑇)‘𝐴) ∈ ℋ) | ||
| Theorem | adj1 31991 | Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih (𝑇‘𝐵)) = (((adjℎ‘𝑇)‘𝐴) ·ih 𝐵)) | ||
| Theorem | adj2 31992 | Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝑇‘𝐴) ·ih 𝐵) = (𝐴 ·ih ((adjℎ‘𝑇)‘𝐵))) | ||
| Theorem | adjeq 31993* | A property that determines the adjoint of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑇‘𝑥) ·ih 𝑦) = (𝑥 ·ih (𝑆‘𝑦))) → (adjℎ‘𝑇) = 𝑆) | ||
| Theorem | adjadj 31994 | Double adjoint. Theorem 3.11(iv) of [Beran] p. 106. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ dom adjℎ → (adjℎ‘(adjℎ‘𝑇)) = 𝑇) | ||
| Theorem | adjvalval 31995* | Value of the value of the adjoint function. (Contributed by NM, 22-Feb-2006.) (Proof shortened by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) |
| ⊢ ((𝑇 ∈ dom adjℎ ∧ 𝐴 ∈ ℋ) → ((adjℎ‘𝑇)‘𝐴) = (℩𝑤 ∈ ℋ ∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·ih 𝐴) = (𝑥 ·ih 𝑤))) | ||
| Theorem | unopadj2 31996 | The adjoint of a unitary operator is its inverse (converse). Equation 2 of [AkhiezerGlazman] p. 72. (Contributed by NM, 23-Feb-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ UniOp → (adjℎ‘𝑇) = ◡𝑇) | ||
| Theorem | hmopadj 31997 | A Hermitian operator is self-adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ HrmOp → (adjℎ‘𝑇) = 𝑇) | ||
| Theorem | hmdmadj 31998 | Every Hermitian operator has an adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ dom adjℎ) | ||
| Theorem | hmopadj2 31999 | An operator is Hermitian iff it is self-adjoint. Definition of Hermitian in [Halmos] p. 41. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ dom adjℎ → (𝑇 ∈ HrmOp ↔ (adjℎ‘𝑇) = 𝑇)) | ||
| Theorem | hmoplin 32000 | A Hermitian operator is linear. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) |
| ⊢ (𝑇 ∈ HrmOp → 𝑇 ∈ LinOp) | ||
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