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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | hilmetdval 28901 | 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 28902 | 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 28903 | 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 28904 | 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 28905* | Lemma used for derivation of the completeness axiom ax-hcompl 28907 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 28906* | Lemma used for derivation of the completeness axiom ax-hcompl 28907 from ZFC Hilbert space theory. The first five hypotheses would be satisfied by the definitions described in ax-hilex 28704; the 6th would be satisfied by eqid 2821; the 7th by a given fixed Hilbert space; and the last by theorem hlcompl 28620. (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 28907* | Completeness of a Hilbert space. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.) |
⊢ (𝐹 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥) | ||
Theorem | hhcms 28908 | 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 28909 | 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 28910 | 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 28911 | 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 28912 | Define the set of subspaces of a Hilbert space. See issh 28913 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 28913 | 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 28914* | 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 28915 | 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 28916 | 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 28917 | 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 28918 | 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 28919 | 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 28920 | 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 28921 | 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 28922 | 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 28923 | 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 28924* | 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 28925 | 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 28926 | 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 28927. From Definition of [Beran] p. 107. Alternate definitions are given by isch2 28928 and isch3 28946. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.) |
⊢ Cℋ = {ℎ ∈ Sℋ ∣ ( ⇝𝑣 “ (ℎ ↑m ℕ)) ⊆ ℎ} | ||
Theorem | isch 28927 | 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 28928* | 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 28929 | 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 28930 | 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 28931 | 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 28932 | A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
⊢ 𝐻 ∈ Cℋ ⇒ ⊢ 𝐻 ∈ Sℋ | ||
Theorem | ch0 28933 | 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 28934 | 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 28935 | 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 28936 | 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 28937 | 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 28938 | 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 28939 | 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 28940 | 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 28941 | 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 28942 | Function-like behavior of the convergence relation. (Contributed by Mario Carneiro, 14-May-2014.) (New usage is discouraged.) |
⊢ ⇝𝑣 :dom ⇝𝑣 ⟶ ℋ | ||
Theorem | hlimuni 28943 | 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 28944* | 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 28945* | 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 28946* | 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 28947* | 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 28948 | The unit Hilbert lattice element (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 28949 | 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 28950 | Hilbert space is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.) |
⊢ ℋ ∈ Sℋ | ||
Theorem | shsspwh 28951 | Subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.) |
⊢ Sℋ ⊆ 𝒫 ℋ | ||
Theorem | chsspwh 28952 | Closed subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.) |
⊢ Cℋ ⊆ 𝒫 ℋ | ||
Theorem | hsn0elch 28953 | 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 28954 | 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 28955* | 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 28956 | 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 28957* | 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 28985 and chocvali 29004 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 28958 | Define the zero for closed subspaces of Hilbert space. See h0elch 28960 for closure law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) |
⊢ 0ℋ = {0ℎ} | ||
Theorem | elch0 28959 | 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 28960 | 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 28961 | The zero subspace is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.) |
⊢ 0ℋ ∈ Sℋ | ||
Theorem | hhssva 28962 | 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 28963 | 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 28964 | 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 28965* | Lemma for hhssabloilem 28966. (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 28966 | Lemma for hhssabloi 28967. Formerly part of proof for hhssabloi 28967 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 28967 | 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 28968 | Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.) |
⊢ (𝐻 ∈ Sℋ → ( +ℎ ↾ (𝐻 × 𝐻)) ∈ AbelOp) | ||
Theorem | hhssnv 28969 | 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 28970 | 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 28971 | 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 28972 | Lemma for hhsssh 28974. (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 28973 | Lemma for hhsssh 28974. (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 28974 | 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 28975 | 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 28976 | 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 28977 | 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 28978 | 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 28979 | 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 28980 | 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 28981 | 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 28982 | 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 28983 | 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 28984 | Obsolete version of cssbn 23907: 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 28985* | 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 28986* | 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 28987* | 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 28988 | 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 28989 | 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 ) | ||
Theorem | ocss 28990 | An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.) |
⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ) | ||
Theorem | shocss 28991 | An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.) |
⊢ (𝐴 ∈ Sℋ → (⊥‘𝐴) ⊆ ℋ) | ||
Theorem | occon 28992 | Contraposition law for orthogonal complement. (Contributed by NM, 8-Aug-2000.) (New usage is discouraged.) |
⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ⊆ 𝐵 → (⊥‘𝐵) ⊆ (⊥‘𝐴))) | ||
Theorem | occon2 28993 | Double contraposition for orthogonal complement. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.) |
⊢ ((𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ) → (𝐴 ⊆ 𝐵 → (⊥‘(⊥‘𝐴)) ⊆ (⊥‘(⊥‘𝐵)))) | ||
Theorem | occon2i 28994 | Double contraposition for orthogonal complement. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.) |
⊢ 𝐴 ⊆ ℋ & ⊢ 𝐵 ⊆ ℋ ⇒ ⊢ (𝐴 ⊆ 𝐵 → (⊥‘(⊥‘𝐴)) ⊆ (⊥‘(⊥‘𝐵))) | ||
Theorem | oc0 28995 | The zero vector belongs to an orthogonal complement of a Hilbert subspace. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.) |
⊢ (𝐻 ∈ Sℋ → 0ℎ ∈ (⊥‘𝐻)) | ||
Theorem | ocorth 28996 | Members of a subset and its complement are orthogonal. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.) |
⊢ (𝐻 ⊆ ℋ → ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ (⊥‘𝐻)) → (𝐴 ·ih 𝐵) = 0)) | ||
Theorem | shocorth 28997 | Members of a subspace and its complement are orthogonal. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.) |
⊢ (𝐻 ∈ Sℋ → ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ (⊥‘𝐻)) → (𝐴 ·ih 𝐵) = 0)) | ||
Theorem | ococss 28998 | Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.) |
⊢ (𝐴 ⊆ ℋ → 𝐴 ⊆ (⊥‘(⊥‘𝐴))) | ||
Theorem | shococss 28999 | Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.) |
⊢ (𝐴 ∈ Sℋ → 𝐴 ⊆ (⊥‘(⊥‘𝐴))) | ||
Theorem | shorth 29000 | Members of orthogonal subspaces are orthogonal. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.) |
⊢ (𝐻 ∈ Sℋ → (𝐺 ⊆ (⊥‘𝐻) → ((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐻) → (𝐴 ·ih 𝐵) = 0))) |
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