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Type | Label | Description |
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Statement | ||
Theorem | hcauseq 30301 | 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 30302* | 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 30303* | 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 30304* | 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 30305 | 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 30306 | 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 30307* | 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 30308* | 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 30309* | 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 30310 | 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 30311 | 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 30312 | 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 30313 | 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 30314 | 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 30315 | 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 30316* | Lemma used for derivation of the completeness axiom ax-hcompl 30318 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 30317* | Lemma used for derivation of the completeness axiom ax-hcompl 30318 from ZFC Hilbert space theory. The first five hypotheses would be satisfied by the definitions described in ax-hilex 30115; the 6th would be satisfied by eqid 2731; the 7th by a given fixed Hilbert space; and the last by Theorem hlcompl 30031. (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 30318* | Completeness of a Hilbert space. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.) |
⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥) | ||
Theorem | hhcms 30319 | 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 30320 | 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 30321 | 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 30322 | 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 30323 | Define the set of subspaces of a Hilbert space. See issh 30324 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 30324 | 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 30325* | 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 30326 | 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 30327 | 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 30328 | 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 30329 | 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 30330 | 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 30331 | 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 30332 | 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 30333 | 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 30334 | 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 30335* | 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 30336 | 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 30337 | 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 30338. From Definition of [Beran] p. 107. Alternate definitions are given by isch2 30339 and isch3 30357. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.) |
⊢ Cℋ = {ℎ ∈ Sℋ ∣ ( ⇝𝑣 “ (ℎ ↑m ℕ)) ⊆ ℎ} | ||
Theorem | isch 30338 | 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 30339* | 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 30340 | 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 30341 | 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 30342 | 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 30343 | A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ ⇒ ⊢ 𝐻 ∈ Sℋ | ||
Theorem | ch0 30344 | 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 30345 | 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 30346 | 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 30347 | 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 30348 | 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 30349 | 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 30350 | 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 30351 | 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 30352 | 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 30353 | Function-like behavior of the convergence relation. (Contributed by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
⊢ ⇝𝑣 :dom ⇝𝑣 ⟶ ℋ | ||
Theorem | hlimuni 30354 | 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 30355* | 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 30356* | 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 30357* | 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 30358* | 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 30359 | 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 30360 | 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 30361 | Hilbert space is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.) |
⊢ ℋ ∈ Sℋ | ||
Theorem | shsspwh 30362 | Subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.) |
⊢ Sℋ ⊆ 𝒫 ℋ | ||
Theorem | chsspwh 30363 | Closed subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.) |
⊢ Cℋ ⊆ 𝒫 ℋ | ||
Theorem | hsn0elch 30364 | 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 30365 | 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 30366* | 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 30367 | 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 30368* | 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 30396 and chocvali 30415 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 30369 | Define the zero for closed subspaces of Hilbert space. See h0elch 30371 for closure law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) |
⊢ 0ℋ = {0ℎ} | ||
Theorem | elch0 30370 | 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 30371 | 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 30372 | The zero subspace is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.) |
⊢ 0ℋ ∈ Sℋ | ||
Theorem | hhssva 30373 | 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 30374 | 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 30375 | 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 30376* | Lemma for hhssabloilem 30377. (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 30377 | Lemma for hhssabloi 30378. Formerly part of proof for hhssabloi 30378 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 30378 | 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 30379 | Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.) |
⊢ (𝐻 ∈ Sℋ → ( +ℎ ↾ (𝐻 × 𝐻)) ∈ AbelOp) | ||
Theorem | hhssnv 30380 | 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 30381 | 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 30382 | 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 30383 | Lemma for hhsssh 30385. (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 30384 | Lemma for hhsssh 30385. (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 30385 | 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 30386 | 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 30387 | 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 30388 | 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 30389 | 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 | ||
Theorem | hhssims 30390 | Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) |
⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 & ⊢ 𝐻 ∈ Sℋ & ⊢ 𝐷 = ((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻)) ⇒ ⊢ 𝐷 = (IndMet‘𝑊) | ||
Theorem | hhssims2 30391 | Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) |
⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 & ⊢ 𝐷 = (IndMet‘𝑊) & ⊢ 𝐻 ∈ Sℋ ⇒ ⊢ 𝐷 = ((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻)) | ||
Theorem | hhssmet 30392 | Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) |
⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 & ⊢ 𝐷 = (IndMet‘𝑊) & ⊢ 𝐻 ∈ Sℋ ⇒ ⊢ 𝐷 ∈ (Met‘𝐻) | ||
Theorem | hhssmetdval 30393 | Value of the distance function of the metric space of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) |
⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 & ⊢ 𝐷 = (IndMet‘𝑊) & ⊢ 𝐻 ∈ Sℋ ⇒ ⊢ ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) → (𝐴𝐷𝐵) = (normℎ‘(𝐴 −ℎ 𝐵))) | ||
Theorem | hhsscms 30394 | The induced metric of a closed subspace is complete. (Contributed by NM, 10-Apr-2008.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 & ⊢ 𝐷 = (IndMet‘𝑊) & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ 𝐷 ∈ (CMet‘𝐻) | ||
Theorem | hhssbnOLD 30395 | Obsolete version of cssbn 24821: Banach space property of a closed subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 & ⊢ 𝐻 ∈ Cℋ ⇒ ⊢ 𝑊 ∈ CBan | ||
Theorem | ocval 30396* | Value of orthogonal complement of a subset of Hilbert space. (Contributed by NM, 7-Aug-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
⊢ (𝐻 ⊆ ℋ → (⊥‘𝐻) = {𝑥 ∈ ℋ ∣ ∀𝑦 ∈ 𝐻 (𝑥 ·ih 𝑦) = 0}) | ||
Theorem | ocel 30397* | Membership in orthogonal complement of H subset. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.) |
⊢ (𝐻 ⊆ ℋ → (𝐴 ∈ (⊥‘𝐻) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ 𝐻 (𝐴 ·ih 𝑥) = 0))) | ||
Theorem | shocel 30398* | Membership in orthogonal complement of H subspace. (Contributed by NM, 9-Oct-1999.) (New usage is discouraged.) |
⊢ (𝐻 ∈ Sℋ → (𝐴 ∈ (⊥‘𝐻) ↔ (𝐴 ∈ ℋ ∧ ∀𝑥 ∈ 𝐻 (𝐴 ·ih 𝑥) = 0))) | ||
Theorem | ocsh 30399 | The orthogonal complement of a subspace is a subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.) |
⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ∈ Sℋ ) | ||
Theorem | shocsh 30400 | The orthogonal complement of a subspace is a subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.) |
⊢ (𝐴 ∈ Sℋ → (⊥‘𝐴) ∈ Sℋ ) |
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