HomeTheorem List (p. 314 of 504)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | polid 31301 | Polarization identity. Recovers inner product from norm. Exercise 4(a) of [ReedSimon] p. 63. The outermost operation is + instead of - due to our mathematicians' (rather than physicists') version of Axiom ax-his3 31226. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (((((normℎ‘(𝐴 +ℎ 𝐵))↑2) − ((normℎ‘(𝐴 −ℎ 𝐵))↑2)) + (i · (((normℎ‘(𝐴 +ℎ (i ·ℎ 𝐵)))↑2) − ((normℎ‘(𝐴 −ℎ (i ·ℎ 𝐵)))↑2)))) / 4)) | ||
| Theorem | hilablo 31302 | Hilbert space vector addition is an Abelian group operation. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.) |
| ⊢ +ℎ ∈ AbelOp | ||
| Theorem | hilid 31303 | The group identity element of Hilbert space vector addition is the zero vector. (Contributed by NM, 16-Apr-2007.) (New usage is discouraged.) |
| ⊢ (GId‘ +ℎ ) = 0ℎ | ||
| Theorem | hilvc 31304 | Hilbert space is a complex vector space. Vector addition is +ℎ, and scalar product is ·ℎ. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.) |
| ⊢ ⟨ +ℎ , ·ℎ ⟩ ∈ CVecOLD | ||
| Theorem | hilnormi 31305 | Hilbert space norm in terms of vector space norm. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| ⊢ ℋ = (BaseSet‘𝑈) & ⊢ ·ih = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ NrmCVec ⇒ ⊢ normℎ = (normCV‘𝑈) | ||
| Theorem | hilhhi 31306 | Deduce the structure of Hilbert space from its components. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) |
| ⊢ ℋ = (BaseSet‘𝑈) & ⊢ +ℎ = ( +𝑣 ‘𝑈) & ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) & ⊢ ·ih = (·𝑖OLD‘𝑈) & ⊢ 𝑈 ∈ NrmCVec ⇒ ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ | ||
| Theorem | hhnv 31307 | Hilbert space is a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ ⇒ ⊢ 𝑈 ∈ NrmCVec | ||
| Theorem | hhva 31308 | The group (addition) operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ ⇒ ⊢ +ℎ = ( +𝑣 ‘𝑈) | ||
| Theorem | hhba 31309 | The base set of Hilbert space. This theorem provides an independent proof of df-hba 31111 (see comments in that definition). (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ ⇒ ⊢ ℋ = (BaseSet‘𝑈) | ||
| Theorem | hh0v 31310 | The zero vector of Hilbert space. (Contributed by NM, 17-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ ⇒ ⊢ 0ℎ = (0vec‘𝑈) | ||
| Theorem | hhsm 31311 | The scalar product operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ ⇒ ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) | ||
| Theorem | hhvs 31312 | The vector subtraction operation of Hilbert space. (Contributed by NM, 13-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ ⇒ ⊢ −ℎ = ( −𝑣 ‘𝑈) | ||
| Theorem | hhnm 31313 | The norm function of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ ⇒ ⊢ normℎ = (normCV‘𝑈) | ||
| Theorem | hhims 31314 | The induced metric of Hilbert space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝐷 = (normℎ ∘ −ℎ ) ⇒ ⊢ 𝐷 = (IndMet‘𝑈) | ||
| Theorem | hhims2 31315 | Hilbert space distance metric. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ 𝐷 = (normℎ ∘ −ℎ ) | ||
| Theorem | hhmet 31316 | The induced metric of Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ 𝐷 ∈ (Met‘ ℋ) | ||
| Theorem | hhxmet 31317 | The induced metric of Hilbert space. (Contributed by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ 𝐷 ∈ (∞Met‘ ℋ) | ||
| Theorem | hhmetdval 31318 | Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴𝐷𝐵) = (normℎ‘(𝐴 −ℎ 𝐵))) | ||
| Theorem | hhip 31319 | The inner product operation of Hilbert space. (Contributed by NM, 17-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ ⇒ ⊢ ·ih = (·𝑖OLD‘𝑈) | ||
| Theorem | hhph 31320 | The Hilbert space of the Hilbert Space Explorer is an inner product space. (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ ⇒ ⊢ 𝑈 ∈ CPreHilOLD | ||
| Theorem | bcsiALT 31321 | Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (abs‘(𝐴 ·ih 𝐵)) ≤ ((normℎ‘𝐴) · (normℎ‘𝐵)) | ||
| Theorem | bcsiHIL 31322 | Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Proved from ZFC version.) (Contributed by NM, 24-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ ℋ & ⊢ 𝐵 ∈ ℋ ⇒ ⊢ (abs‘(𝐴 ·ih 𝐵)) ≤ ((normℎ‘𝐴) · (normℎ‘𝐵)) | ||
| Theorem | bcs 31323 | Bunjakovaskij-Cauchy-Schwarz inequality. Remark 3.4 of [Beran] p. 98. (Contributed by NM, 16-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (abs‘(𝐴 ·ih 𝐵)) ≤ ((normℎ‘𝐴) · (normℎ‘𝐵))) | ||
| Theorem | bcs2 31324 | Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 31322. (Contributed by NM, 24-May-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ (normℎ‘𝐴) ≤ 1) → (abs‘(𝐴 ·ih 𝐵)) ≤ (normℎ‘𝐵)) | ||
| Theorem | bcs3 31325 | Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 31322. (Contributed by NM, 26-May-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ (normℎ‘𝐵) ≤ 1) → (abs‘(𝐴 ·ih 𝐵)) ≤ (normℎ‘𝐴)) | ||
| Theorem | hcau 31326* | Member of the set of Cauchy sequences on a Hilbert space. Definition for Cauchy sequence in [Beran] p. 96. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
| ⊢ (𝐹 ∈ Cauchy ↔ (𝐹:ℕ⟶ ℋ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑦) −ℎ (𝐹‘𝑧))) < 𝑥)) | ||
| Theorem | hcauseq 31327 | A Cauchy sequences on a Hilbert space is a sequence. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
| ⊢ (𝐹 ∈ Cauchy → 𝐹:ℕ⟶ ℋ) | ||
| Theorem | hcaucvg 31328* | A Cauchy sequence on a Hilbert space converges. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
| ⊢ ((𝐹 ∈ Cauchy ∧ 𝐴 ∈ ℝ+) → ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑦) −ℎ (𝐹‘𝑧))) < 𝐴) | ||
| Theorem | seq1hcau 31329* | A sequence on a Hilbert space is a Cauchy sequence if it converges. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
| ⊢ (𝐹:ℕ⟶ ℋ → (𝐹 ∈ Cauchy ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑦) −ℎ (𝐹‘𝑧))) < 𝑥)) | ||
| Theorem | hlimi 31330* | Express the predicate: The limit of vector sequence 𝐹 in a Hilbert space is 𝐴, i.e. 𝐹 converges to 𝐴. This means that for any real 𝑥, no matter how small, there always exists an integer 𝑦 such that the norm of any later vector in the sequence minus the limit is less than 𝑥. Definition of converge in [Beran] p. 96. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ (𝐹 ⇝𝑣 𝐴 ↔ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥)) | ||
| Theorem | hlimseqi 31331 | A sequence with a limit on a Hilbert space is a sequence. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐹:ℕ⟶ ℋ) | ||
| Theorem | hlimveci 31332 | Closure of the limit of a sequence on Hilbert space. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐴 ∈ ℋ) | ||
| Theorem | hlimconvi 31333* | Convergence of a sequence on a Hilbert space. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ ((𝐹 ⇝𝑣 𝐴 ∧ 𝐵 ∈ ℝ+) → ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝐵) | ||
| Theorem | hlim2 31334* | The limit of a sequence on a Hilbert space. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
| ⊢ ((𝐹:ℕ⟶ ℋ ∧ 𝐴 ∈ ℋ) → (𝐹 ⇝𝑣 𝐴 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ ∀𝑧 ∈ (ℤ≥‘𝑦)(normℎ‘((𝐹‘𝑧) −ℎ 𝐴)) < 𝑥)) | ||
| Theorem | hlimadd 31335* | Limit of the sum of two sequences in a Hilbert vector space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) |
| ⊢ (𝜑 → 𝐹:ℕ⟶ ℋ) & ⊢ (𝜑 → 𝐺:ℕ⟶ ℋ) & ⊢ (𝜑 → 𝐹 ⇝𝑣 𝐴) & ⊢ (𝜑 → 𝐺 ⇝𝑣 𝐵) & ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛) +ℎ (𝐺‘𝑛))) ⇒ ⊢ (𝜑 → 𝐻 ⇝𝑣 (𝐴 +ℎ 𝐵)) | ||
| Theorem | hilmet 31336 | The Hilbert space norm determines a metric space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝐷 = (normℎ ∘ −ℎ ) ⇒ ⊢ 𝐷 ∈ (Met‘ ℋ) | ||
| Theorem | hilxmet 31337 | The Hilbert space norm determines a metric space. (Contributed by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) |
| ⊢ 𝐷 = (normℎ ∘ −ℎ ) ⇒ ⊢ 𝐷 ∈ (∞Met‘ ℋ) | ||
| Theorem | hilmetdval 31338 | Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝐷 = (normℎ ∘ −ℎ ) ⇒ ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴𝐷𝐵) = (normℎ‘(𝐴 −ℎ 𝐵))) | ||
| Theorem | hilims 31339 | Hilbert space distance metric. (Contributed by NM, 13-Sep-2007.) (New usage is discouraged.) |
| ⊢ ℋ = (BaseSet‘𝑈) & ⊢ +ℎ = ( +𝑣 ‘𝑈) & ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) & ⊢ ·ih = (·𝑖OLD‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝑈 ∈ NrmCVec ⇒ ⊢ 𝐷 = (normℎ ∘ −ℎ ) | ||
| Theorem | hhcau 31340 | The Cauchy sequences of Hilbert space. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ Cauchy = ((Cau‘𝐷) ∩ ( ℋ ↑m ℕ)) | ||
| Theorem | hhlm 31341 | The limit sequences of Hilbert space. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ ⇝𝑣 = ((⇝𝑡‘𝐽) ↾ ( ℋ ↑m ℕ)) | ||
| Theorem | hhcmpl 31342* | Lemma used for derivation of the completeness axiom ax-hcompl 31344 from ZFC Hilbert space theory. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝐹 ∈ (Cau‘𝐷) → ∃𝑥 ∈ ℋ 𝐹(⇝𝑡‘𝐽)𝑥) ⇒ ⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥) | ||
| Theorem | hilcompl 31343* | Lemma used for derivation of the completeness axiom ax-hcompl 31344 from ZFC Hilbert space theory. The first five hypotheses would be satisfied by the definitions described in ax-hilex 31141; the 6th would be satisfied by eqid 2756; the 7th by a given fixed Hilbert space; and the last by Theorem hlcompl 31057. (Contributed by NM, 13-Sep-2007.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
| ⊢ ℋ = (BaseSet‘𝑈) & ⊢ +ℎ = ( +𝑣 ‘𝑈) & ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) & ⊢ ·ih = (·𝑖OLD‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝑈 ∈ CHilOLD & ⊢ (𝐹 ∈ (Cau‘𝐷) → ∃𝑥 ∈ ℋ 𝐹(⇝𝑡‘𝐽)𝑥) ⇒ ⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥) | ||
| Axiom | ax-hcompl 31344* | Completeness of a Hilbert space. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.) |
| ⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥) | ||
| Theorem | hhcms 31345 | The Hilbert space induced metric determines a complete metric space. (Contributed by NM, 10-Apr-2008.) (Proof shortened by Mario Carneiro, 14-May-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ 𝐷 ∈ (CMet‘ ℋ) | ||
| Theorem | hhhl 31346 | The Hilbert space structure is a complex Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ ⇒ ⊢ 𝑈 ∈ CHilOLD | ||
| Theorem | hilcms 31347 | The Hilbert space norm determines a complete metric space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝐷 = (normℎ ∘ −ℎ ) ⇒ ⊢ 𝐷 ∈ (CMet‘ ℋ) | ||
| Theorem | hilhl 31348 | The Hilbert space of the Hilbert Space Explorer is a complex Hilbert space. (Contributed by Steve Rodriguez, 29-Apr-2007.) (New usage is discouraged.) |
| ⊢ ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ ∈ CHilOLD | ||
| Definition | df-sh 31349 | Define the set of subspaces of a Hilbert space. See issh 31350 for its membership relation. Basically, a subspace is a subset of a Hilbert space that acts like a vector space. From Definition of [Beran] p. 95. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| ⊢ Sℋ = {ℎ ∈ 𝒫 ℋ ∣ (0ℎ ∈ ℎ ∧ ( +ℎ “ (ℎ × ℎ)) ⊆ ℎ ∧ ( ·ℎ “ (ℂ × ℎ)) ⊆ ℎ)} | ||
| Theorem | issh 31350 | Subspace 𝐻 of a Hilbert space. A subspace is a subset of Hilbert space which contains the zero vector and is closed under vector addition and scalar multiplication. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| ⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻))) | ||
| Theorem | issh2 31351* | Subspace 𝐻 of a Hilbert space. A subspace is a subset of Hilbert space which contains the zero vector and is closed under vector addition and scalar multiplication. Definition of [Beran] p. 95. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| ⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻))) | ||
| Theorem | shss 31352 | A subspace is a subset of Hilbert space. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ) | ||
| Theorem | shel 31353 | A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.) |
| ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ) | ||
| Theorem | shex 31354 | The set of subspaces of a Hilbert space exists (is a set). (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.) |
| ⊢ Sℋ ∈ V | ||
| Theorem | shssii 31355 | A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Sℋ ⇒ ⊢ 𝐻 ⊆ ℋ | ||
| Theorem | sheli 31356 | A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Sℋ ⇒ ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ) | ||
| Theorem | shelii 31357 | A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Sℋ & ⊢ 𝐴 ∈ 𝐻 ⇒ ⊢ 𝐴 ∈ ℋ | ||
| Theorem | sh0 31358 | The zero vector belongs to any subspace of a Hilbert space. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| ⊢ (𝐻 ∈ Sℋ → 0ℎ ∈ 𝐻) | ||
| Theorem | shaddcl 31359 | Closure of vector addition in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (New usage is discouraged.) |
| ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) → (𝐴 +ℎ 𝐵) ∈ 𝐻) | ||
| Theorem | shmulcl 31360 | Closure of vector scalar multiplication in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (New usage is discouraged.) |
| ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝐻) → (𝐴 ·ℎ 𝐵) ∈ 𝐻) | ||
| Theorem | issh3 31361* | Subspace 𝐻 of a Hilbert space. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.) |
| ⊢ (𝐻 ⊆ ℋ → (𝐻 ∈ Sℋ ↔ (0ℎ ∈ 𝐻 ∧ (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻)))) | ||
| Theorem | shsubcl 31362 | Closure of vector subtraction in a subspace of a Hilbert space. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.) |
| ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) → (𝐴 −ℎ 𝐵) ∈ 𝐻) | ||
| Definition | df-ch 31363 | Define the set of closed subspaces of a Hilbert space. A closed subspace is one in which the limit of every convergent sequence in the subspace belongs to the subspace. For its membership relation, see isch 31364. From Definition of [Beran] p. 107. Alternate definitions are given by isch2 31365 and isch3 31383. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.) |
| ⊢ Cℋ = {ℎ ∈ Sℋ ∣ ( ⇝𝑣 “ (ℎ ↑m ℕ)) ⊆ ℎ} | ||
| Theorem | isch 31364 | Closed subspace 𝐻 of a Hilbert space. (Contributed by NM, 17-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| ⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ( ⇝𝑣 “ (𝐻 ↑m ℕ)) ⊆ 𝐻)) | ||
| Theorem | isch2 31365* | Closed subspace 𝐻 of a Hilbert space. Definition of [Beran] p. 107. (Contributed by NM, 17-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| ⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ∀𝑓∀𝑥((𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 𝑥) → 𝑥 ∈ 𝐻))) | ||
| Theorem | chsh 31366 | A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| ⊢ (𝐻 ∈ Cℋ → 𝐻 ∈ Sℋ ) | ||
| Theorem | chsssh 31367 | Closed subspaces are subspaces in a Hilbert space. (Contributed by NM, 29-May-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| ⊢ Cℋ ⊆ Sℋ | ||
| Theorem | chex 31368 | The set of closed subspaces of a Hilbert space exists (is a set). (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.) |
| ⊢ Cℋ ∈ V | ||
| Theorem | chshii 31369 | A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ 𝐻 ∈ Sℋ | ||
| Theorem | ch0 31370 | The zero vector belongs to any closed subspace of a Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.) |
| ⊢ (𝐻 ∈ Cℋ → 0ℎ ∈ 𝐻) | ||
| Theorem | chss 31371 | A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.) |
| ⊢ (𝐻 ∈ Cℋ → 𝐻 ⊆ ℋ) | ||
| Theorem | chel 31372 | A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.) |
| ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ) | ||
| Theorem | chssii 31373 | A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ 𝐻 ⊆ ℋ | ||
| Theorem | cheli 31374 | A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ) | ||
| Theorem | chelii 31375 | A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Cℋ & ⊢ 𝐴 ∈ 𝐻 ⇒ ⊢ 𝐴 ∈ ℋ | ||
| Theorem | chlimi 31376 | The limit property of a closed subspace of a Hilbert space. (Contributed by NM, 14-Sep-1999.) (New usage is discouraged.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ ((𝐻 ∈ Cℋ ∧ 𝐹:ℕ⟶𝐻 ∧ 𝐹 ⇝𝑣 𝐴) → 𝐴 ∈ 𝐻) | ||
| Theorem | hlim0 31377 | The zero sequence in Hilbert space converges to the zero vector. (Contributed by NM, 17-Aug-1999.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
| ⊢ (ℕ × {0ℎ}) ⇝𝑣 0ℎ | ||
| Theorem | hlimcaui 31378 | If a sequence in Hilbert space subset converges to a limit, it is a Cauchy sequence. (Contributed by NM, 17-Aug-1999.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
| ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐹 ∈ Cauchy) | ||
| Theorem | hlimf 31379 | Function-like behavior of the convergence relation. (Contributed by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
| ⊢ ⇝𝑣 :dom ⇝𝑣 ⟶ ℋ | ||
| Theorem | hlimuni 31380 | A Hilbert space sequence converges to at most one limit. (Contributed by NM, 19-Aug-1999.) (Revised by Mario Carneiro, 2-May-2015.) (New usage is discouraged.) |
| ⊢ ((𝐹 ⇝𝑣 𝐴 ∧ 𝐹 ⇝𝑣 𝐵) → 𝐴 = 𝐵) | ||
| Theorem | hlimreui 31381* | The limit of a Hilbert space sequence is unique. (Contributed by NM, 19-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
| ⊢ (∃𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥 ↔ ∃!𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥) | ||
| Theorem | hlimeui 31382* | The limit of a Hilbert space sequence is unique. (Contributed by NM, 19-Aug-1999.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
| ⊢ (∃𝑥 𝐹 ⇝𝑣 𝑥 ↔ ∃!𝑥 𝐹 ⇝𝑣 𝑥) | ||
| Theorem | isch3 31383* | A Hilbert subspace is closed iff it is complete. A complete subspace is one in which every Cauchy sequence of vectors in the subspace converges to a member of the subspace (Definition of complete subspace in [Beran] p. 96). Remark 3.12 of [Beran] p. 107. (Contributed by NM, 24-Dec-2001.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
| ⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ∀𝑓 ∈ Cauchy (𝑓:ℕ⟶𝐻 → ∃𝑥 ∈ 𝐻 𝑓 ⇝𝑣 𝑥))) | ||
| Theorem | chcompl 31384* | Completeness of a closed subspace of Hilbert space. (Contributed by NM, 4-Oct-1999.) (New usage is discouraged.) |
| ⊢ ((𝐻 ∈ Cℋ ∧ 𝐹 ∈ Cauchy ∧ 𝐹:ℕ⟶𝐻) → ∃𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥) | ||
| Theorem | helch 31385 | The Hilbert lattice one (which is all of Hilbert space) belongs to the Hilbert lattice. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 6-Sep-1999.) (New usage is discouraged.) |
| ⊢ ℋ ∈ Cℋ | ||
| Theorem | ifchhv 31386 | Prove if(𝐴 ∈ Cℋ , 𝐴, ℋ) ∈ Cℋ. (Contributed by David A. Wheeler, 8-Dec-2018.) (New usage is discouraged.) |
| ⊢ if(𝐴 ∈ Cℋ , 𝐴, ℋ) ∈ Cℋ | ||
| Theorem | helsh 31387 | Hilbert space is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.) |
| ⊢ ℋ ∈ Sℋ | ||
| Theorem | shsspwh 31388 | Subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.) |
| ⊢ Sℋ ⊆ 𝒫 ℋ | ||
| Theorem | chsspwh 31389 | Closed subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.) |
| ⊢ Cℋ ⊆ 𝒫 ℋ | ||
| Theorem | hsn0elch 31390 | The zero subspace belongs to the set of closed subspaces of Hilbert space. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.) |
| ⊢ {0ℎ} ∈ Cℋ | ||
| Theorem | norm1 31391 | From any nonzero Hilbert space vector, construct a vector whose norm is 1. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (normℎ‘((1 / (normℎ‘𝐴)) ·ℎ 𝐴)) = 1) | ||
| Theorem | norm1exi 31392* | A normalized vector exists in a subspace iff the subspace has a nonzero vector. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Sℋ ⇒ ⊢ (∃𝑥 ∈ 𝐻 𝑥 ≠ 0ℎ ↔ ∃𝑦 ∈ 𝐻 (normℎ‘𝑦) = 1) | ||
| Theorem | norm1hex 31393 | A normalized vector can exist only iff the Hilbert space has a nonzero vector. (Contributed by NM, 21-Jan-2006.) (New usage is discouraged.) |
| ⊢ (∃𝑥 ∈ ℋ 𝑥 ≠ 0ℎ ↔ ∃𝑦 ∈ ℋ (normℎ‘𝑦) = 1) | ||
| Definition | df-oc 31394* | Define orthogonal complement of a subset (usually a subspace) of Hilbert space. The orthogonal complement is the set of all vectors orthogonal to all vectors in the subset. See ocval 31422 and chocvali 31441 for its value. Textbooks usually denote this unary operation with the symbol ⊥ as a small superscript, although Mittelstaedt uses the symbol as a prefix operation. Here we define a function (prefix operation) ⊥ rather than introducing a new syntactic form. This lets us take advantage of the theorems about functions that we already have proved under set theory. Definition of [Mittelstaedt] p. 9. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.) |
| ⊢ ⊥ = (𝑥 ∈ 𝒫 ℋ ↦ {𝑦 ∈ ℋ ∣ ∀𝑧 ∈ 𝑥 (𝑦 ·ih 𝑧) = 0}) | ||
| Definition | df-ch0 31395 | Define the zero for closed subspaces of Hilbert space. See h0elch 31397 for closure law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) |
| ⊢ 0ℋ = {0ℎ} | ||
| Theorem | elch0 31396 | Membership in zero for closed subspaces of Hilbert space. (Contributed by NM, 6-Apr-2001.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ) | ||
| Theorem | h0elch 31397 | The zero subspace is a closed subspace. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) |
| ⊢ 0ℋ ∈ Cℋ | ||
| Theorem | h0elsh 31398 | The zero subspace is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.) |
| ⊢ 0ℋ ∈ Sℋ | ||
| Theorem | hhssva 31399 | The vector addition operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.) |
| ⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩ ⇒ ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) = ( +𝑣 ‘𝑊) | ||
| Theorem | hhsssm 31400 | The scalar multiplication operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.) |
| ⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩ ⇒ ⊢ ( ·ℎ ↾ (ℂ × 𝐻)) = ( ·𝑠OLD ‘𝑊) | ||
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