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Theorem List for Metamath Proof Explorer - 22701-22800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremshsspwh 22701 Subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)

Theoremchsspwh 22702 Closed subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.)

Theoremhsn0elch 22703 The zero subspace belongs to the set of closed subspaces of Hilbert space. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)

Theoremnorm1 22704 From any nonzero Hilbert space vector, construct a vector whose norm is 1. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)

Theoremnorm1exi 22705* 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 22706 A normalized vector can exist only iff the Hilbert space has a nonzero vector. (Contributed by NM, 21-Jan-2006.) (New usage is discouraged.)

18.4.3  Orthocomplements

Definitiondf-oc 22707* 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 22735 and chocvali 22754 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 22708 Define the zero for closed subspaces of Hilbert space. See h0elch 22710 for closure law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)

Theoremelch0 22709 Membership in zero for closed subspaces of Hilbert space. (Contributed by NM, 6-Apr-2001.) (New usage is discouraged.)

Theoremh0elch 22710 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 22711 The zero subspace is a subspace of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)

Theoremhhssva 22712 The vector addition operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)

Theoremhhsssm 22713 The scalar multiplication operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)

Theoremhhssnm 22714 The norm operation on a subspace. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
CV

Theoremhhssabloi 22715 Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremhhssablo 22716 Abelian group property of subspace addition. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.)

Theoremhhssnv 22717 Normed complex vector space property of a subspace. (Contributed by NM, 26-Mar-2008.) (New usage is discouraged.)

Theoremhhssnvt 22718 Normed complex vector space property of a subspace. (Contributed by NM, 9-Apr-2008.) (New usage is discouraged.)

Theoremhhsst 22719 A member of is a subspace. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)

Theoremhhshsslem1 22720 Lemma for hhsssh 22722. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhshsslem2 22721 Lemma for hhsssh 22722. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)

Theoremhhsssh 22722 The predicate " is a subspace of Hilbert space." (Contributed by NM, 25-Mar-2008.) (New usage is discouraged.)

Theoremhhsssh2 22723 The predicate " is a subspace of Hilbert space." (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)

Theoremhhssba 22724 The base set of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhssvs 22725 The vector subtraction operation on a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhssvsf 22726 Mapping of the vector subtraction operation on a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhssph 22727 Inner product space property of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhssims 22728 Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhssims2 22729 Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhssmet 22730 Induced metric of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhssmetdval 22731 Value of the distance function of the metric space of a subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhsscms 22732 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 22733 Banach space property of a closed subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremhhsshl 22734 Hilbert space property of a closed subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremocval 22735* 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 22736* Membership in orthogonal complement of H subset. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)

Theoremshocel 22737* Membership in orthogonal complement of H subspace. (Contributed by NM, 9-Oct-1999.) (New usage is discouraged.)

Theoremocsh 22738 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 22739 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 22740 An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)

Theoremshocss 22741 An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)

Theoremoccon 22742 Contraposition law for orthogonal complement. (Contributed by NM, 8-Aug-2000.) (New usage is discouraged.)

Theoremoccon2 22743 Double contraposition for orthogonal complement. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)

Theoremoccon2i 22744 Double contraposition for orthogonal complement. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)

Theoremoc0 22745 The zero vector belongs to an orthogonal complement of a Hilbert subspace. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)

Theoremocorth 22746 Members of a subset and its complement are orthogonal. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)

Theoremshocorth 22747 Members of a subspace and its complement are orthogonal. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)

Theoremococss 22748 Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)

Theoremshococss 22749 Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)

Theoremshorth 22750 Members of orthogonal subspaces are orthogonal. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)

Theoremocin 22751 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 22752 Hilbert lattice contraposition law. (Contributed by Mario Carneiro, 18-May-2014.) (New usage is discouraged.)

Theoremocnel 22753 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 22754* 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 22755 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 22756 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 22757* 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 22758 Lemma for occl 22759. (Contributed by NM, 7-Aug-2000.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)

Theoremoccl 22759 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 22760 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 22761 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 22762 Closure of orthocomplement. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)

18.4.4  Subspace sum, span, lattice join, lattice supremum

Definitiondf-shs 22763* Define subspace sum in . See shsval 22767, shsval2i 22842, and shsval3i 22843 for its value. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)

Definitiondf-span 22764* Define the linear span of a subset of Hilbert space. Definition of span in [Schechter] p. 276. See spanval 22788 for its value. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)

Definitiondf-chj 22765* Define Hilbert lattice join. See chjval 22807 for its value and chjcl 22812 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 22810. (Contributed by NM, 1-Nov-2000.) (New usage is discouraged.)

Definitiondf-chsup 22766 Define the supremum of a set of Hilbert lattice elements. See chsupval2 22865 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 22794. (Contributed by NM, 9-Dec-2003.) (New usage is discouraged.)

Theoremshsval 22767 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 22768 The subspace sum is a subset of Hilbert space. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremshsel 22769* 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 22770* Membership in the subspace sum of two Hilbert subspaces, using vector subtraction. (Contributed by NM, 20-Jan-2007.) (New usage is discouraged.)

Theoremshseli 22771* Membership in subspace sum. (Contributed by NM, 4-May-2000.) (New usage is discouraged.)

Theoremshscli 22772 Closure of subspace sum. (Contributed by NM, 15-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremshscl 22773 Closure of subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)

Theoremshscom 22774 Commutative law for subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)

Theoremshsva 22775 Vector sum belongs to subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)

Theoremshsel1 22776 A subspace sum contains a member of one of its subspaces. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)

Theoremshsel2 22777 A subspace sum contains a member of one of its subspaces. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)

Theoremshsvs 22778 Vector subtraction belongs to subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)

Theoremshsub1 22779 Subspace sum is an upper bound of its arguments. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.)

Theoremshsub2 22780 Subspace sum is an upper bound of its arguments. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)

Theoremchoc0 22781 The orthocomplement of the zero subspace is the unit subspace. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchoc1 22782 The orthocomplement of the unit subspace is the zero subspace. Does not require Axiom of Choice. (Contributed by NM, 24-Oct-1999.) (New usage is discouraged.)

Theoremchocnul 22783 Orthogonal complement of the empty set. (Contributed by NM, 31-Oct-2000.) (New usage is discouraged.)

Theoremshintcli 22784 Closure of intersection of a non-empty subset of . (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)

Theoremshintcl 22785 The intersection of a non-empty set of subspaces is a subspace. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)

Theoremchintcli 22786 The intersection of a non-empty set of closed subspaces is a closed subspace. (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)

Theoremchintcl 22787 The intersection (infimum) of a non-empty subset of belongs to . Part of Theorem 3.13 of [Beran] p. 108. Also part of Definition 3.4-1 in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)

Theoremspanval 22788* Value of the linear span of a subset of Hilbert space. The span is the intersection of all subspaces constraining the subset. Definition of span in [Schechter] p. 276. (Contributed by NM, 2-Jun-2004.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremhsupval 22789 Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 22864. (Contributed by NM, 9-Dec-2003.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremchsupval 22790 The value of the supremum of a set of closed subspaces of Hilbert space. For an alternate version of the value, see chsupval2 22865. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)

Theoremspancl 22791 The span of a subset of Hilbert space is a subspace. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)

Theoremelspancl 22792 A member of a span is a vector. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)

Theoremshsupcl 22793 Closure of the subspace supremum of set of subsets of Hilbert space. (Contributed by NM, 26-Nov-2004.) (New usage is discouraged.)

Theoremhsupcl 22794 Closure of supremum of set of subsets of Hilbert space. Note that the supremum belongs to even if the subsets do not. (Contributed by NM, 10-Nov-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremchsupcl 22795 Closure of supremum of subset of . Definition of supremum in Proposition 1 of [Kalmbach] p. 65. Shows that is a complete lattice. Also part of Definition 3.4-1 in [MegPav2000] p. 2345 (PDF p. 8). (Contributed by NM, 10-Nov-1999.) (New usage is discouraged.)

Theoremhsupss 22796 Subset relation for supremum of Hilbert space subsets. (Contributed by NM, 24-Nov-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremchsupss 22797 Subset relation for supremum of subset of . (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)

Theoremhsupunss 22798 The union of a set of Hilbert space subsets is smaller than its supremum. (Contributed by NM, 24-Nov-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremchsupunss 22799 The union of a set of closed subspaces is smaller than its supremum. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)

Theoremspanss2 22800 A subset of Hilbert space is included in its span. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)

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