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Theorem List for Metamath Proof Explorer - 21601-21700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhlim2 21601* 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.)

Theoremhlimadd 21602* Limit of the sum of two sequences in a Hilbert vector space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)

15.9.12  Derivation of the completeness axiom from ZF set theory

Theoremhilmet 21603 The Hilbert space norm determines a metric space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)

Theoremhilxmet 21604 The Hilbert space norm determines a metric space. (Contributed by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)

Theoremhilmetdval 21605 Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)

Theoremhilims 21606 Hilbert space distance metric. (Contributed by NM, 13-Sep-2007.) (New usage is discouraged.)

Theoremhhcau 21607 The Cauchy sequences of Hilbert space. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.)

Theoremhhlm 21608 The limit sequences of Hilbert space. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.)

Theoremhhcmpl 21609* Lemma used for derivation of the completeness axiom ax-hcompl 21611 from ZFC Hilbert space theory. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.)

Theoremhilcompl 21610* Lemma used for derivation of the completeness axiom ax-hcompl 21611 from ZFC Hilbert space theory. The first 5 hypotheses would be satisfied by the definitions described in ax-hilex 21409; the 6th would be satisfied by eqid 2253; the 7th by a given fixed Hilbert space; and the last by theorem hlcompl 21324. (Contributed by NM, 13-Sep-2007.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)

15.9.13  Completeness postulate for a Hilbert space

Axiomax-hcompl 21611* Completeness of a Hilbert space. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)

15.9.14  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces

Theoremhhcms 21612 The Hilbert space induced metric determines a complete metric space. (Contributed by NM, 10-Apr-2008.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)

Theoremhhhl 21613 The Hilbert space structure is a complex Hilbert space. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhilcms 21614 The Hilbert space norm determines a complete metric space. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)

Theoremhilhl 21615 The Hilbert space of the Hilbert Space Explorer is a complex Hilbert space. (Contributed by Steve Rodriguez, 29-Apr-2007.) (New usage is discouraged.)

15.9.15  Subspaces

Definitiondf-sh 21616 Define the set of subspaces of a Hilbert space. See issh 21617 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.)

Theoremissh 21617 Subspace 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.)

Theoremissh2 21618* Subspace 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.)

Theoremshss 21619 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.)

Theoremshel 21620 A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.)

Theoremshex 21621 The set of subspaces of a Hilbert space exists (is a set). (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)

Theoremshssii 21622 A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)

Theoremsheli 21623 A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)

Theoremshelii 21624 A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)

Theoremsh0 21625 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.)

Theoremshaddcl 21626 Closure of vector addition in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (New usage is discouraged.)

Theoremshmulcl 21627 Closure of vector scalar multiplication in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (New usage is discouraged.)

TheoremshmulclOLD 21628 Closure of vector scalar multiplication in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremissh3 21629* Subspace of a Hilbert space. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)

Theoremshsubcl 21630 Closure of vector subtraction in a subspace of a Hilbert space. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)

15.9.16  Closed subspaces

Definitiondf-ch 21631 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 21632. From Definition of [Beran] p. 107. Alternate definitions are given by isch2 21633 and isch3 21651. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)

Theoremisch 21632 Closed subspace of a Hilbert space. (Contributed by NM, 17-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremisch2 21633* Closed subspace 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.)

Theoremchsh 21634 A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremchsssh 21635 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.)

Theoremchex 21636 The set of closed subspaces of a Hilbert space exists (is a set). (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)

Theoremchshii 21637 A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Theoremch0 21638 The zero vector belongs to any closed subspace of a Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.)

Theoremchss 21639 A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.)

Theoremchel 21640 A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)

Theoremchssii 21641 A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)

Theoremcheli 21642 A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)

Theoremchelii 21643 A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)

Theoremchlimi 21644 The limit property of a closed subspace of a Hilbert space. (Contributed by NM, 14-Sep-1999.) (New usage is discouraged.)

Theoremhlim0 21645 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.)

Theoremhlimcaui 21646 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.)

Theoremhlimf 21647 Function-like behavior of the convergence relation. (Contributed by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)

Theoremhlimuni 21648 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.)

Theoremhlimreui 21649* 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.)

Theoremhlimeui 21650* 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.)

Theoremisch3 21651* 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.)

Theoremchcompl 21652* Completeness of a closed subspace of Hilbert space. (Contributed by NM, 4-Oct-1999.) (New usage is discouraged.)

Theoremhelch 21653 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.)

Theoremhelsh 21654 Hilbert space is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)

Theoremshsspwh 21655 Subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)

Theoremchsspwh 21656 Closed subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)

Theoremhsn0elch 21657 The zero subspace belongs to the set of closed subspaces of Hilbert space. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)

Theoremnorm1 21658 From any nonzero Hilbert space vector, construct a vector whose norm is 1. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)

Theoremnorm1exi 21659* A normalized vector exists in a subspace iff the subspace has a nonzero vector. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.)

Theoremnorm1hex 21660 A normalized vector can exist only iff the Hilbert space has a nonzero vector. (Contributed by NM, 21-Jan-2006.) (New usage is discouraged.)

15.9.17  Orthocomplements

Definitiondf-oc 21661* 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 21689 and chocvali 21708 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 syntactical 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.)

Definitiondf-ch0 21662 Define the zero for closed subspaces of Hilbert space. See h0elch 21664 for closure law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)

Theoremelch0 21663 Membership in zero for closed subspaces of Hilbert space. (Contributed by NM, 6-Apr-2001.) (New usage is discouraged.)

Theoremh0elch 21664 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.)

Theoremh0elsh 21665 The zero subspace is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)

Theoremhhssva 21666 The vector addition operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)

Theoremhhsssm 21667 The scalar multiplication operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)

Theoremhhssnm 21668 The norm operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
CV

Theoremhhssabloi 21669 Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremhhssablo 21670 Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.)

Theoremhhssnv 21671 Normed complex vector space property of a subspace. (Contributed by NM, 26-Mar-2008.) (New usage is discouraged.)

Theoremhhssnvt 21672 Normed complex vector space property of a subspace. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.)

Theoremhhsst 21673 A member of is a subspace. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)

Theoremhhshsslem1 21674 Lemma for hhsssh 21676. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhshsslem2 21675 Lemma for hhsssh 21676. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)

Theoremhhsssh 21676 The predicate " is a subspace of Hilbert space." (Contributed by NM, 25-Mar-2008.) (New usage is discouraged.)

Theoremhhsssh2 21677 The predicate " is a subspace of Hilbert space." (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)

Theoremhhssba 21678 The base set of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhssvs 21679 The vector subtraction operation on a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhssvsf 21680 Mapping of the vector subtraction operation on a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhssph 21681 Inner product space property of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhssims 21682 Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhssims2 21683 Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhssmet 21684 Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhssmetdval 21685 Value of the distance function of the metric space of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhsscms 21686 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.)

Theoremhhssbn 21687 Banach space property of a closed subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhsshl 21688 Hilbert space property of a closed subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremocval 21689* 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.)

Theoremocel 21690* Membership in orthogonal complement of H subset. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)

Theoremshocel 21691* Membership in orthogonal complement of H subspace. (Contributed by NM, 9-Oct-1999.) (New usage is discouraged.)

Theoremocsh 21692 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.)

Theoremshocsh 21693 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.)

Theoremocss 21694 An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)

Theoremshocss 21695 An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)

Theoremoccon 21696 Contraposition law for orthogonal complement. (Contributed by NM, 8-Aug-2000.) (New usage is discouraged.)

Theoremoccon2 21697 Double contraposition for orthogonal complement. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)

Theoremoccon2i 21698 Double contraposition for orthogonal complement. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)

Theoremoc0 21699 The zero vector belongs to an orthogonal complement of a Hilbert subspace. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)

Theoremocorth 21700 Members of a subset and its complement are orthogonal. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)

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