HomeTheorem List (p. 312 of 496)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | hhmet 31101 | 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 31102 | 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 31103 | 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 31104 | 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 31105 | 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 31106 | 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 31107 | 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 31108 | 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 31109 | Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 31107. (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 31110 | Corollary of the Bunjakovaskij-Cauchy-Schwarz inequality bcsiHIL 31107. (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 31111* | 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 31112 | 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 31113* | 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 31114* | 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 31115* | 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 31116 | 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 31117 | 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 31118* | 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 31119* | 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 31120* | 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 31121 | 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 31122 | 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 31123 | 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 31124 | 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 31125 | 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 31126 | 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 31127* | Lemma used for derivation of the completeness axiom ax-hcompl 31129 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 31128* | Lemma used for derivation of the completeness axiom ax-hcompl 31129 from ZFC Hilbert space theory. The first five hypotheses would be satisfied by the definitions described in ax-hilex 30926; the 6th would be satisfied by eqid 2735; the 7th by a given fixed Hilbert space; and the last by Theorem hlcompl 30842. (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 31129* | Completeness of a Hilbert space. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.) |
| ⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥) | ||
| Theorem | hhcms 31130 | 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 31131 | 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 31132 | 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 31133 | 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 31134 | Define the set of subspaces of a Hilbert space. See issh 31135 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 31135 | 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 31136* | 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 31137 | 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 31138 | 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 31139 | 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 31140 | 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 31141 | 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 31142 | 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 31143 | 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 31144 | 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 31145 | 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 31146* | 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 31147 | 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 31148 | 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 31149. From Definition of [Beran] p. 107. Alternate definitions are given by isch2 31150 and isch3 31168. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.) |
| ⊢ Cℋ = {ℎ ∈ Sℋ ∣ ( ⇝𝑣 “ (ℎ ↑m ℕ)) ⊆ ℎ} | ||
| Theorem | isch 31149 | 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 31150* | 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 31151 | 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 31152 | 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 31153 | 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 31154 | A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ 𝐻 ∈ Sℋ | ||
| Theorem | ch0 31155 | 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 31156 | 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 31157 | 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 31158 | 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 31159 | 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 31160 | 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 31161 | 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 31162 | 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 31163 | 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 31164 | Function-like behavior of the convergence relation. (Contributed by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
| ⊢ ⇝𝑣 :dom ⇝𝑣 ⟶ ℋ | ||
| Theorem | hlimuni 31165 | 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 31166* | 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 31167* | 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 31168* | 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 31169* | 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 31170 | 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 31171 | 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 31172 | Hilbert space is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.) |
| ⊢ ℋ ∈ Sℋ | ||
| Theorem | shsspwh 31173 | Subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.) |
| ⊢ Sℋ ⊆ 𝒫 ℋ | ||
| Theorem | chsspwh 31174 | Closed subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.) |
| ⊢ Cℋ ⊆ 𝒫 ℋ | ||
| Theorem | hsn0elch 31175 | 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 31176 | 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 31177* | 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 31178 | 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 31179* | 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 31207 and chocvali 31226 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 31180 | Define the zero for closed subspaces of Hilbert space. See h0elch 31182 for closure law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) |
| ⊢ 0ℋ = {0ℎ} | ||
| Theorem | elch0 31181 | 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 31182 | 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 31183 | The zero subspace is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.) |
| ⊢ 0ℋ ∈ Sℋ | ||
| Theorem | hhssva 31184 | 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 31185 | 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) | ||
| Theorem | hhssnm 31186 | The norm operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.) |
| ⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩ ⇒ ⊢ (normℎ ↾ 𝐻) = (normCV‘𝑊) | ||
| Theorem | issubgoilem 31187* | Lemma for hhssabloilem 31188. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.) |
| ⊢ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) → (𝑥𝐻𝑦) = (𝑥𝐺𝑦)) ⇒ ⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌) → (𝐴𝐻𝐵) = (𝐴𝐺𝐵)) | ||
| Theorem | hhssabloilem 31188 | Lemma for hhssabloi 31189. Formerly part of proof for hhssabloi 31189 which was based on the deprecated definition "SubGrpOp" for subgroups. (Contributed by NM, 9-Apr-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (Revised by AV, 27-Aug-2021.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Sℋ ⇒ ⊢ ( +ℎ ∈ GrpOp ∧ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ GrpOp ∧ ( +ℎ ↾ (𝐻 × 𝐻)) ⊆ +ℎ ) | ||
| Theorem | hhssabloi 31189 | Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (Proof shortened by AV, 27-Aug-2021.) (New usage is discouraged.) |
| ⊢ 𝐻 ∈ Sℋ ⇒ ⊢ ( +ℎ ↾ (𝐻 × 𝐻)) ∈ AbelOp | ||
| Theorem | hhssablo 31190 | Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.) |
| ⊢ (𝐻 ∈ Sℋ → ( +ℎ ↾ (𝐻 × 𝐻)) ∈ AbelOp) | ||
| Theorem | hhssnv 31191 | Normed complex vector space property of a subspace. (Contributed by NM, 26-Mar-2008.) (New usage is discouraged.) |
| ⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩ & ⊢ 𝐻 ∈ Sℋ ⇒ ⊢ 𝑊 ∈ NrmCVec | ||
| Theorem | hhssnvt 31192 | Normed complex vector space property of a subspace. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.) |
| ⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩ ⇒ ⊢ (𝐻 ∈ Sℋ → 𝑊 ∈ NrmCVec) | ||
| Theorem | hhsst 31193 | A member of Sℋ is a subspace. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩ ⇒ ⊢ (𝐻 ∈ Sℋ → 𝑊 ∈ (SubSp‘𝑈)) | ||
| Theorem | hhshsslem1 31194 | Lemma for hhsssh 31196. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩ & ⊢ 𝑊 ∈ (SubSp‘𝑈) & ⊢ 𝐻 ⊆ ℋ ⇒ ⊢ 𝐻 = (BaseSet‘𝑊) | ||
| Theorem | hhshsslem2 31195 | Lemma for hhsssh 31196. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩ & ⊢ 𝑊 ∈ (SubSp‘𝑈) & ⊢ 𝐻 ⊆ ℋ ⇒ ⊢ 𝐻 ∈ Sℋ | ||
| Theorem | hhsssh 31196 | The predicate "𝐻 is a subspace of Hilbert space." (Contributed by NM, 25-Mar-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ & ⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩ ⇒ ⊢ (𝐻 ∈ Sℋ ↔ (𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ)) | ||
| Theorem | hhsssh2 31197 | The predicate "𝐻 is a subspace of Hilbert space." (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.) |
| ⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩ ⇒ ⊢ (𝐻 ∈ Sℋ ↔ (𝑊 ∈ NrmCVec ∧ 𝐻 ⊆ ℋ)) | ||
| Theorem | hhssba 31198 | The base set of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) |
| ⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩ & ⊢ 𝐻 ∈ Sℋ ⇒ ⊢ 𝐻 = (BaseSet‘𝑊) | ||
| Theorem | hhssvs 31199 | The vector subtraction operation on a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) |
| ⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩ & ⊢ 𝐻 ∈ Sℋ ⇒ ⊢ ( −ℎ ↾ (𝐻 × 𝐻)) = ( −𝑣 ‘𝑊) | ||
| Theorem | hhssvsf 31200 | Mapping of the vector subtraction operation on a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) |
| ⊢ 𝑊 = ⟨⟨( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))⟩, (normℎ ↾ 𝐻)⟩ & ⊢ 𝐻 ∈ Sℋ ⇒ ⊢ ( −ℎ ↾ (𝐻 × 𝐻)):(𝐻 × 𝐻)⟶𝐻 | ||
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