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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | hlcompl 29001 | Completeness of a Hilbert space. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 9-May-2014.) (New usage is discouraged.) |
⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝐹 ∈ dom (⇝𝑡‘𝐽)) | ||
Theorem | cnchl 29002 | The set of complex numbers is a complex Hilbert space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 + , · 〉, abs〉 ⇒ ⊢ 𝑈 ∈ CHilOLD | ||
Theorem | htthlem 29003* | Lemma for htth 29004. The collection 𝐾, which consists of functions 𝐹(𝑧)(𝑤) = 〈𝑤 ∣ 𝑇(𝑧)〉 = 〈𝑇(𝑤) ∣ 𝑧〉 for each 𝑧 in the unit ball, is a collection of bounded linear functions by ipblnfi 28941, so by the Uniform Boundedness theorem ubth 28959, there is a uniform bound 𝑦 on ∥ 𝐹(𝑥) ∥ for all 𝑥 in the unit ball. Then ∣ 𝑇(𝑥) ∣ ↑2 = 〈𝑇(𝑥) ∣ 𝑇(𝑥)〉 = 𝐹(𝑥)( 𝑇(𝑥)) ≤ 𝑦 ∣ 𝑇(𝑥) ∣, so ∣ 𝑇(𝑥) ∣ ≤ 𝑦 and 𝑇 is bounded. (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝐿 = (𝑈 LnOp 𝑈) & ⊢ 𝐵 = (𝑈 BLnOp 𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑈 ∈ CHilOLD & ⊢ 𝑊 = 〈〈 + , · 〉, abs〉 & ⊢ (𝜑 → 𝑇 ∈ 𝐿) & ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑃(𝑇‘𝑦)) = ((𝑇‘𝑥)𝑃𝑦)) & ⊢ 𝐹 = (𝑧 ∈ 𝑋 ↦ (𝑤 ∈ 𝑋 ↦ (𝑤𝑃(𝑇‘𝑧)))) & ⊢ 𝐾 = (𝐹 “ {𝑧 ∈ 𝑋 ∣ (𝑁‘𝑧) ≤ 1}) ⇒ ⊢ (𝜑 → 𝑇 ∈ 𝐵) | ||
Theorem | htth 29004* | Hellinger-Toeplitz Theorem: any self-adjoint linear operator defined on all of Hilbert space is bounded. Theorem 10.1-1 of [Kreyszig] p. 525. Discovered by E. Hellinger and O. Toeplitz in 1910, "it aroused both admiration and puzzlement since the theorem establishes a relation between properties of two different kinds, namely, the properties of being defined everywhere and being bounded." (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝐿 = (𝑈 LnOp 𝑈) & ⊢ 𝐵 = (𝑈 BLnOp 𝑈) ⇒ ⊢ ((𝑈 ∈ CHilOLD ∧ 𝑇 ∈ 𝐿 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑃(𝑇‘𝑦)) = ((𝑇‘𝑥)𝑃𝑦)) → 𝑇 ∈ 𝐵) | ||
This part contains the definitions and theorems used by the Hilbert Space Explorer (HSE), mmhil.html. Because it axiomatizes a single complex Hilbert space whose existence is assumed, its usefulness is limited. For example, it cannot work with real or quaternion Hilbert spaces and it cannot study relationships between two Hilbert spaces. More information can be found on the Hilbert Space Explorer page. Future development should instead work with general Hilbert spaces as defined by df-hil 20671; note that df-hil 20671 uses extensible structures. The intent is for this deprecated section to be deleted once all its theorems have been translated into extensible structure versions (or are not useful). Many of the theorems in this section have already been translated to extensible structure versions, but there is still a lot in this section that might be useful for future reference. It is much easier to translate these by hand from this section than to start from scratch from textbook proofs, since the HSE omits no details. | ||
Syntax | chba 29005 | Extend class notation with Hilbert vector space. |
class ℋ | ||
Syntax | cva 29006 | Extend class notation with vector addition in Hilbert space. In the literature, the subscript "h" is omitted, but we need it to avoid ambiguity with complex number addition + caddc 10737. |
class +ℎ | ||
Syntax | csm 29007 | Extend class notation with scalar multiplication in Hilbert space. In the literature scalar multiplication is usually indicated by juxtaposition, but we need an explicit symbol to prevent ambiguity. |
class ·ℎ | ||
Syntax | csp 29008 | Extend class notation with inner (scalar) product in Hilbert space. In the literature, the inner product of 𝐴 and 𝐵 is usually written 〈𝐴, 𝐵〉 but our operation notation allows us to use existing theorems about operations and also eliminates ambiguity with the definition of an ordered pair df-op 4553. |
class ·ih | ||
Syntax | cno 29009 | Extend class notation with the norm function in Hilbert space. In the literature, the norm of 𝐴 is usually written "|| 𝐴 ||", but we use function notation to take advantage of our existing theorems about functions. |
class normℎ | ||
Syntax | c0v 29010 | Extend class notation with zero vector in Hilbert space. |
class 0ℎ | ||
Syntax | cmv 29011 | Extend class notation with vector subtraction in Hilbert space. |
class −ℎ | ||
Syntax | ccauold 29012 | Extend class notation with set of Cauchy sequences in Hilbert space. |
class Cauchy | ||
Syntax | chli 29013 | Extend class notation with convergence relation in Hilbert space. |
class ⇝𝑣 | ||
Syntax | csh 29014 | Extend class notation with set of subspaces of a Hilbert space. |
class Sℋ | ||
Syntax | cch 29015 | Extend class notation with set of closed subspaces of a Hilbert space. |
class Cℋ | ||
Syntax | cort 29016 | Extend class notation with orthogonal complement in Cℋ. |
class ⊥ | ||
Syntax | cph 29017 | Extend class notation with subspace sum in Cℋ. |
class +ℋ | ||
Syntax | cspn 29018 | Extend class notation with subspace span in Cℋ. |
class span | ||
Syntax | chj 29019 | Extend class notation with join in Cℋ. |
class ∨ℋ | ||
Syntax | chsup 29020 | Extend class notation with supremum of a collection in Cℋ. |
class ∨ℋ | ||
Syntax | c0h 29021 | Extend class notation with zero of Cℋ. |
class 0ℋ | ||
Syntax | ccm 29022 | Extend class notation with the commutes relation on a Hilbert lattice. |
class 𝐶ℋ | ||
Syntax | cpjh 29023 | Extend class notation with set of projections on a Hilbert space. |
class projℎ | ||
Syntax | chos 29024 | Extend class notation with sum of Hilbert space operators. |
class +op | ||
Syntax | chot 29025 | Extend class notation with scalar product of a Hilbert space operator. |
class ·op | ||
Syntax | chod 29026 | Extend class notation with difference of Hilbert space operators. |
class −op | ||
Syntax | chfs 29027 | Extend class notation with sum of Hilbert space functionals. |
class +fn | ||
Syntax | chft 29028 | Extend class notation with scalar product of Hilbert space functional. |
class ·fn | ||
Syntax | ch0o 29029 | Extend class notation with the Hilbert space zero operator. |
class 0hop | ||
Syntax | chio 29030 | Extend class notation with Hilbert space identity operator. |
class Iop | ||
Syntax | cnop 29031 | Extend class notation with the operator norm function. |
class normop | ||
Syntax | ccop 29032 | Extend class notation with set of continuous Hilbert space operators. |
class ContOp | ||
Syntax | clo 29033 | Extend class notation with set of linear Hilbert space operators. |
class LinOp | ||
Syntax | cbo 29034 | Extend class notation with set of bounded linear operators. |
class BndLinOp | ||
Syntax | cuo 29035 | Extend class notation with set of unitary Hilbert space operators. |
class UniOp | ||
Syntax | cho 29036 | Extend class notation with set of Hermitian Hilbert space operators. |
class HrmOp | ||
Syntax | cnmf 29037 | Extend class notation with the functional norm function. |
class normfn | ||
Syntax | cnl 29038 | Extend class notation with the functional nullspace function. |
class null | ||
Syntax | ccnfn 29039 | Extend class notation with set of continuous Hilbert space functionals. |
class ContFn | ||
Syntax | clf 29040 | Extend class notation with set of linear Hilbert space functionals. |
class LinFn | ||
Syntax | cado 29041 | Extend class notation with Hilbert space adjoint function. |
class adjℎ | ||
Syntax | cbr 29042 | Extend class notation with the bra of a vector in Dirac bra-ket notation. |
class bra | ||
Syntax | ck 29043 | Extend class notation with the outer product of two vectors in Dirac bra-ket notation. |
class ketbra | ||
Syntax | cleo 29044 | Extend class notation with positive operator ordering. |
class ≤op | ||
Syntax | cei 29045 | Extend class notation with Hilbert space eigenvector function. |
class eigvec | ||
Syntax | cel 29046 | Extend class notation with Hilbert space eigenvalue function. |
class eigval | ||
Syntax | cspc 29047 | Extend class notation with the spectrum of an operator. |
class Lambda | ||
Syntax | cst 29048 | Extend class notation with set of states on a Hilbert lattice. |
class States | ||
Syntax | chst 29049 | Extend class notation with set of Hilbert-space-valued states on a Hilbert lattice. |
class CHStates | ||
Syntax | ccv 29050 | Extend class notation with the covers relation on a Hilbert lattice. |
class ⋖ℋ | ||
Syntax | cat 29051 | Extend class notation with set of atoms on a Hilbert lattice. |
class HAtoms | ||
Syntax | cmd 29052 | Extend class notation with the modular pair relation on a Hilbert lattice. |
class 𝑀ℋ | ||
Syntax | cdmd 29053 | Extend class notation with the dual modular pair relation on a Hilbert lattice. |
class 𝑀ℋ* | ||
Definition | df-hnorm 29054 | Define the function for the norm of a vector of Hilbert space. See normval 29210 for its value and normcl 29211 for its closure. Theorems norm-i-i 29219, norm-ii-i 29223, and norm-iii-i 29225 show it has the expected properties of a norm. In the literature, the norm of 𝐴 is usually written "|| 𝐴 ||", but we use function notation to take advantage of our existing theorems about functions. Definition of norm in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.) |
⊢ normℎ = (𝑥 ∈ dom dom ·ih ↦ (√‘(𝑥 ·ih 𝑥))) | ||
Definition | df-hba 29055 | Define base set of Hilbert space, for use if we want to develop Hilbert space independently from the axioms (see comments in ax-hilex 29085). Note that ℋ is considered a primitive in the Hilbert space axioms below, and we don't use this definition outside of this section. This definition can be proved independently from those axioms as Theorem hhba 29253. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
⊢ ℋ = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | ||
Definition | df-h0v 29056 | Define the zero vector of Hilbert space. Note that 0vec is considered a primitive in the Hilbert space axioms below, and we don't use this definition outside of this section. It is proved from the axioms as Theorem hh0v 29254. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
⊢ 0ℎ = (0vec‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | ||
Definition | df-hvsub 29057* | Define vector subtraction. See hvsubvali 29106 for its value and hvsubcli 29107 for its closure. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.) |
⊢ −ℎ = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑥 +ℎ (-1 ·ℎ 𝑦))) | ||
Definition | df-hlim 29058* | Define the limit relation for Hilbert space. See hlimi 29274 for its relational expression. Note that 𝑓:ℕ⟶ ℋ is an infinite sequence of vectors, i.e. a mapping from integers to vectors. Definition of converge in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.) |
⊢ ⇝𝑣 = {〈𝑓, 𝑤〉 ∣ ((𝑓:ℕ⟶ ℋ ∧ 𝑤 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑧) −ℎ 𝑤)) < 𝑥)} | ||
Definition | df-hcau 29059* | Define the set of Cauchy sequences on a Hilbert space. See hcau 29270 for its membership relation. Note that 𝑓:ℕ⟶ ℋ is an infinite sequence of vectors, i.e. a mapping from integers to vectors. Definition of Cauchy sequence in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.) |
⊢ Cauchy = {𝑓 ∈ ( ℋ ↑m ℕ) ∣ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝑓‘𝑦) −ℎ (𝑓‘𝑧))) < 𝑥} | ||
Theorem | h2hva 29060 | The group (addition) operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑈 ∈ NrmCVec ⇒ ⊢ +ℎ = ( +𝑣 ‘𝑈) | ||
Theorem | h2hsm 29061 | The scalar product operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑈 ∈ NrmCVec ⇒ ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) | ||
Theorem | h2hnm 29062 | The norm function of Hilbert space. (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑈 ∈ NrmCVec ⇒ ⊢ normℎ = (normCV‘𝑈) | ||
Theorem | h2hvs 29063 | The vector subtraction operation of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑈 ∈ NrmCVec & ⊢ ℋ = (BaseSet‘𝑈) ⇒ ⊢ −ℎ = ( −𝑣 ‘𝑈) | ||
Theorem | h2hmetdval 29064 | Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑈 ∈ NrmCVec & ⊢ ℋ = (BaseSet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴𝐷𝐵) = (normℎ‘(𝐴 −ℎ 𝐵))) | ||
Theorem | h2hcau 29065 | The Cauchy sequences of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑈 ∈ NrmCVec & ⊢ ℋ = (BaseSet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ Cauchy = ((Cau‘𝐷) ∩ ( ℋ ↑m ℕ)) | ||
Theorem | h2hlm 29066 | The limit sequences of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑈 ∈ NrmCVec & ⊢ ℋ = (BaseSet‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ ⇝𝑣 = ((⇝𝑡‘𝐽) ↾ ( ℋ ↑m ℕ)) | ||
Before introducing the 18 axioms for Hilbert space, we first prove them as the conclusions of Theorems axhilex-zf 29067 through axhcompl-zf 29084, using ZFC set theory only. These show that if we are given a known, fixed Hilbert space 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 that satisfies their hypotheses, then we can derive the Hilbert space axioms as theorems of ZFC set theory. In practice, in order to use these theorems to convert the Hilbert Space explorer to a ZFC-only subtheory, we would also have to provide definitions for the 3 (otherwise primitive) class constants +ℎ, ·ℎ, and ·ih before df-hnorm 29054 above. See also the comment in ax-hilex 29085. | ||
Theorem | axhilex-zf 29067 | Derive Axiom ax-hilex 29085 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ ℋ ∈ V | ||
Theorem | axhfvadd-zf 29068 | Derive Axiom ax-hfvadd 29086 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | ||
Theorem | axhvcom-zf 29069 | Derive Axiom ax-hvcom 29087 from Hilbert space under ZF set theory. (Contributed by NM, 27-May-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴)) | ||
Theorem | axhvass-zf 29070 | Derive Axiom ax-hvass 29088 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶))) | ||
Theorem | axhv0cl-zf 29071 | Derive Axiom ax-hv0cl 29089 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ 0ℎ ∈ ℋ | ||
Theorem | axhvaddid-zf 29072 | Derive Axiom ax-hvaddid 29090 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴) | ||
Theorem | axhfvmul-zf 29073 | Derive Axiom ax-hfvmul 29091 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ ·ℎ :(ℂ × ℋ)⟶ ℋ | ||
Theorem | axhvmulid-zf 29074 | Derive Axiom ax-hvmulid 29092 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴) | ||
Theorem | axhvmulass-zf 29075 | Derive Axiom ax-hvmulass 29093 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ℎ 𝐶) = (𝐴 ·ℎ (𝐵 ·ℎ 𝐶))) | ||
Theorem | axhvdistr1-zf 29076 | Derive Axiom ax-hvdistr1 29094 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 +ℎ 𝐶)) = ((𝐴 ·ℎ 𝐵) +ℎ (𝐴 ·ℎ 𝐶))) | ||
Theorem | axhvdistr2-zf 29077 | Derive Axiom ax-hvdistr2 29095 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) ·ℎ 𝐶) = ((𝐴 ·ℎ 𝐶) +ℎ (𝐵 ·ℎ 𝐶))) | ||
Theorem | axhvmul0-zf 29078 | Derive Axiom ax-hvmul0 29096 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ (𝐴 ∈ ℋ → (0 ·ℎ 𝐴) = 0ℎ) | ||
Theorem | axhfi-zf 29079 | Derive Axiom ax-hfi 29165 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑈 ∈ CHilOLD & ⊢ ·ih = (·𝑖OLD‘𝑈) ⇒ ⊢ ·ih :( ℋ × ℋ)⟶ℂ | ||
Theorem | axhis1-zf 29080 | Derive Axiom ax-his1 29168 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑈 ∈ CHilOLD & ⊢ ·ih = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴))) | ||
Theorem | axhis2-zf 29081 | Derive Axiom ax-his2 29169 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑈 ∈ CHilOLD & ⊢ ·ih = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) ·ih 𝐶) = ((𝐴 ·ih 𝐶) + (𝐵 ·ih 𝐶))) | ||
Theorem | axhis3-zf 29082 | Derive Axiom ax-his3 29170 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑈 ∈ CHilOLD & ⊢ ·ih = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·ℎ 𝐵) ·ih 𝐶) = (𝐴 · (𝐵 ·ih 𝐶))) | ||
Theorem | axhis4-zf 29083 | Derive Axiom ax-his4 29171 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑈 ∈ CHilOLD & ⊢ ·ih = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (𝐴 ·ih 𝐴)) | ||
Theorem | axhcompl-zf 29084* | Derive Axiom ax-hcompl 29288 from Hilbert space under ZF set theory. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) (New usage is discouraged.) |
⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 & ⊢ 𝑈 ∈ CHilOLD ⇒ ⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥) | ||
Here we introduce the axioms a complex Hilbert space, which is the foundation for quantum mechanics and quantum field theory. The 18 axioms for a complex Hilbert space consist of ax-hilex 29085, ax-hfvadd 29086, ax-hvcom 29087, ax-hvass 29088, ax-hv0cl 29089, ax-hvaddid 29090, ax-hfvmul 29091, ax-hvmulid 29092, ax-hvmulass 29093, ax-hvdistr1 29094, ax-hvdistr2 29095, ax-hvmul0 29096, ax-hfi 29165, ax-his1 29168, ax-his2 29169, ax-his3 29170, ax-his4 29171, and ax-hcompl 29288. The axioms specify the properties of 5 primitive symbols, ℋ, +ℎ, ·ℎ, 0ℎ, and ·ih. If we can prove in ZFC set theory that a class 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 is a complex Hilbert space, i.e. that 𝑈 ∈ CHilOLD, then these axioms can be proved as Theorems axhilex-zf 29067, axhfvadd-zf 29068, axhvcom-zf 29069, axhvass-zf 29070, axhv0cl-zf 29071, axhvaddid-zf 29072, axhfvmul-zf 29073, axhvmulid-zf 29074, axhvmulass-zf 29075, axhvdistr1-zf 29076, axhvdistr2-zf 29077, axhvmul0-zf 29078, axhfi-zf 29079, axhis1-zf 29080, axhis2-zf 29081, axhis3-zf 29082, axhis4-zf 29083, and axhcompl-zf 29084 respectively. In that case, the theorems of the Hilbert Space Explorer will become theorems of ZFC set theory. See also the comments in axhilex-zf 29067. | ||
Axiom | ax-hilex 29085 | This is our first axiom for a complex Hilbert space, which is the foundation for quantum mechanics and quantum field theory. We assume that there exists a primitive class, ℋ, which contains objects called vectors. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.) |
⊢ ℋ ∈ V | ||
Axiom | ax-hfvadd 29086 | Vector addition is an operation on ℋ. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.) |
⊢ +ℎ :( ℋ × ℋ)⟶ ℋ | ||
Axiom | ax-hvcom 29087 | Vector addition is commutative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴)) | ||
Axiom | ax-hvass 29088 | Vector addition is associative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶))) | ||
Axiom | ax-hv0cl 29089 | The zero vector is in the vector space. (Contributed by NM, 29-May-1999.) (New usage is discouraged.) |
⊢ 0ℎ ∈ ℋ | ||
Axiom | ax-hvaddid 29090 | Addition with the zero vector. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴) | ||
Axiom | ax-hfvmul 29091 | Scalar multiplication is an operation on ℂ and ℋ. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.) |
⊢ ·ℎ :(ℂ × ℋ)⟶ ℋ | ||
Axiom | ax-hvmulid 29092 | Scalar multiplication by one. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴) | ||
Axiom | ax-hvmulass 29093 | Scalar multiplication associative law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ℎ 𝐶) = (𝐴 ·ℎ (𝐵 ·ℎ 𝐶))) | ||
Axiom | ax-hvdistr1 29094 | Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ (𝐵 +ℎ 𝐶)) = ((𝐴 ·ℎ 𝐵) +ℎ (𝐴 ·ℎ 𝐶))) | ||
Axiom | ax-hvdistr2 29095 | Scalar multiplication distributive law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) ·ℎ 𝐶) = ((𝐴 ·ℎ 𝐶) +ℎ (𝐵 ·ℎ 𝐶))) | ||
Axiom | ax-hvmul0 29096 | Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 29112 and hvsubval 29102). (Contributed by NM, 29-May-1999.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℋ → (0 ·ℎ 𝐴) = 0ℎ) | ||
Theorem | hvmulex 29097 | The Hilbert space scalar product operation is a set. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.) |
⊢ ·ℎ ∈ V | ||
Theorem | hvaddcl 29098 | Closure of vector addition. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) ∈ ℋ) | ||
Theorem | hvmulcl 29099 | Closure of scalar multiplication. (Contributed by NM, 19-Apr-2007.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ℎ 𝐵) ∈ ℋ) | ||
Theorem | hvmulcli 29100 | Closure inference for scalar multiplication. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (𝐴 ·ℎ 𝐵) ∈ ℋ |
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