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17.4.2  Closed subspaces

Definitiondf-ch 21801 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 21802. From Definition of [Beran] p. 107. Alternate definitions are given by isch2 21803 and isch3 21821. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)

Theoremisch 21802 Closed subspace of a Hilbert space. (Contributed by NM, 17-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremisch2 21803* 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 21804 A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremchsssh 21805 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 21806 The set of closed subspaces of a Hilbert space exists (is a set). (Contributed by NM, 23-Oct-1999.) (New usage is discouraged.)

Theoremchshii 21807 A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Theoremch0 21808 The zero vector belongs to any closed subspace of a Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.)

Theoremchss 21809 A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 24-Aug-1999.) (New usage is discouraged.)

Theoremchel 21810 A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)

Theoremchssii 21811 A closed subspace of a Hilbert space is a subset of Hilbert space. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)

Theoremcheli 21812 A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)

Theoremchelii 21813 A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)

Theoremchlimi 21814 The limit property of a closed subspace of a Hilbert space. (Contributed by NM, 14-Sep-1999.) (New usage is discouraged.)

Theoremhlim0 21815 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 21816 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 21817 Function-like behavior of the convergence relation. (Contributed by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)

Theoremhlimuni 21818 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 21819* 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 21820* 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 21821* 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 21822* Completeness of a closed subspace of Hilbert space. (Contributed by NM, 4-Oct-1999.) (New usage is discouraged.)

Theoremhelch 21823 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 21824 Hilbert space is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)

Theoremshsspwh 21825 Subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)

Theoremchsspwh 21826 Closed subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)

Theoremhsn0elch 21827 The zero subspace belongs to the set of closed subspaces of Hilbert space. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)

Theoremnorm1 21828 From any nonzero Hilbert space vector, construct a vector whose norm is 1. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)

Theoremnorm1exi 21829* 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 21830 A normalized vector can exist only iff the Hilbert space has a nonzero vector. (Contributed by NM, 21-Jan-2006.) (New usage is discouraged.)

17.4.3  Orthocomplements

Definitiondf-oc 21831* 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 21859 and chocvali 21878 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 21832 Define the zero for closed subspaces of Hilbert space. See h0elch 21834 for closure law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)

Theoremelch0 21833 Membership in zero for closed subspaces of Hilbert space. (Contributed by NM, 6-Apr-2001.) (New usage is discouraged.)

Theoremh0elch 21834 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 21835 The zero subspace is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)

Theoremhhssva 21836 The vector addition operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)

Theoremhhsssm 21837 The scalar multiplication operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)

Theoremhhssnm 21838 The norm operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
CV

Theoremhhssabloi 21839 Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremhhssablo 21840 Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.)

Theoremhhssnv 21841 Normed complex vector space property of a subspace. (Contributed by NM, 26-Mar-2008.) (New usage is discouraged.)

Theoremhhssnvt 21842 Normed complex vector space property of a subspace. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.)

Theoremhhsst 21843 A member of is a subspace. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)

Theoremhhshsslem1 21844 Lemma for hhsssh 21846. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhshsslem2 21845 Lemma for hhsssh 21846. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)

Theoremhhsssh 21846 The predicate " is a subspace of Hilbert space." (Contributed by NM, 25-Mar-2008.) (New usage is discouraged.)

Theoremhhsssh2 21847 The predicate " is a subspace of Hilbert space." (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)

Theoremhhssba 21848 The base set of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhssvs 21849 The vector subtraction operation on a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhssvsf 21850 Mapping of the vector subtraction operation on a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhssph 21851 Inner product space property of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhssims 21852 Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhssims2 21853 Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhssmet 21854 Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhssmetdval 21855 Value of the distance function of the metric space of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhsscms 21856 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 21857 Banach space property of a closed subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhsshl 21858 Hilbert space property of a closed subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremocval 21859* 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 21860* Membership in orthogonal complement of H subset. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)

Theoremshocel 21861* Membership in orthogonal complement of H subspace. (Contributed by NM, 9-Oct-1999.) (New usage is discouraged.)

Theoremocsh 21862 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 21863 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 21864 An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)

Theoremshocss 21865 An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)

Theoremoccon 21866 Contraposition law for orthogonal complement. (Contributed by NM, 8-Aug-2000.) (New usage is discouraged.)

Theoremoccon2 21867 Double contraposition for orthogonal complement. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)

Theoremoccon2i 21868 Double contraposition for orthogonal complement. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)

Theoremoc0 21869 The zero vector belongs to an orthogonal complement of a Hilbert subspace. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)

Theoremocorth 21870 Members of a subset and its complement are orthogonal. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)

Theoremshocorth 21871 Members of a subspace and its complement are orthogonal. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)

Theoremococss 21872 Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)

Theoremshococss 21873 Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)

Theoremshorth 21874 Members of orthogonal subspaces are orthogonal. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)

Theoremocin 21875 Intersection of a Hilbert subspace and its complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)

Theoremoccon3 21876 Hilbert lattice contraposition law. (Contributed by Mario Carneiro, 18-May-2014.) (New usage is discouraged.)

Theoremocnel 21877 A nonzero vector in the complement of a subspace does not belong to the subspace. (Contributed by NM, 10-Apr-2006.) (New usage is discouraged.)

Theoremchocvali 21878* Value of the orthogonal complement of a Hilbert lattice element. The orthogonal complement of is the set of vectors that are orthogonal to all vectors in . (Contributed by NM, 8-Aug-2004.) (New usage is discouraged.)

Theoremshuni 21879 Two subspaces with trivial intersection have a unique decomposition of the elements of the subspace sum. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremchocunii 21880 Lemma for uniqueness part of Projection Theorem. Theorem 3.7(i) of [Beran] p. 102 (uniqueness part). (Contributed by NM, 23-Oct-1999.) (Proof shortened by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theorempjhthmo 21881* Projection Theorem, uniqueness part. Any two disjoint subspaces yield a unique decomposition of vectors into each subspace. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremoccllem 21882 Lemma for occl 21883. (Contributed by NM, 7-Aug-2000.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)

Theoremoccl 21883 Closure of complement of Hilbert subset. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 8-Aug-2000.) (Proof shortened by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)

Theoremshoccl 21884 Closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 13-Oct-1999.) (New usage is discouraged.)

Theoremchoccl 21885 Closure of complement of Hilbert subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)

Theoremchoccli 21886 Closure of orthocomplement. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)

17.4.4  Subspace sum, span, lattice join, lattice supremum

Definitiondf-shs 21887* Define subspace sum in . See shsval 21891, shsval2i 21966, and shsval3i 21967 for its value. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)

Definitiondf-span 21888* Define the linear span of a subset of Hilbert space. Definition of span in [Schechter] p. 276. See spanval 21912 for its value. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)

Definitiondf-chj 21889* Define Hilbert lattice join. See chjval 21931 for its value and chjcl 21936 for its closure law. Note that we define it over all Hilbert space subsets to allow proving more general theorems. Even for general subsets the join belongs to ; see sshjcl 21934. (Contributed by NM, 1-Nov-2000.) (New usage is discouraged.)

Definitiondf-chsup 21890 Define the supremum of a set of Hilbert lattice elements. See chsupval2 21989 for its value. We actually define the supremum for an arbitrary collection of Hilbert space subsets, not just elements of the Hilbert lattice , to allow more general theorems. Even for general subsets the supremum still a Hilbert lattice element; see hsupcl 21918. (Contributed by NM, 9-Dec-2003.) (New usage is discouraged.)

Theoremshsval 21891 Value of subspace sum of two Hilbert space subspaces. Definition of subspace sum in [Kalmbach] p. 65. (Contributed by NM, 16-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremshsss 21892 The subspace sum is a subset of Hilbert space. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremshsel 21893* Membership in the subspace sum of two Hilbert subspaces. (Contributed by NM, 14-Dec-2004.) (Revised by Mario Carneiro, 29-Jan-2014.) (New usage is discouraged.)

Theoremshsel3 21894* Membership in the subspace sum of two Hilbert subspaces, using vector subtraction. (Contributed by NM, 20-Jan-2007.) (New usage is discouraged.)

Theoremshseli 21895* Membership in subspace sum. (Contributed by NM, 4-May-2000.) (New usage is discouraged.)

Theoremshscli 21896 Closure of subspace sum. (Contributed by NM, 15-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremshscl 21897 Closure of subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)

Theoremshscom 21898 Commutative law for subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)

Theoremshsva 21899 Vector sum belongs to subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)

Theoremshsel1 21900 A subspace sum contains a member of one of its subspaces. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)

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