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

Theoremhhsshl 22701 Hilbert space property of a closed subspace. (Contributed by NM, 10-Apr-2008.) (New usage is discouraged.)

Theoremocval 22702* 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 22703* Membership in orthogonal complement of H subset. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)

Theoremshocel 22704* Membership in orthogonal complement of H subspace. (Contributed by NM, 9-Oct-1999.) (New usage is discouraged.)

Theoremocsh 22705 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 22706 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 22707 An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)

Theoremshocss 22708 An orthogonal complement is a subset of Hilbert space. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)

Theoremoccon 22709 Contraposition law for orthogonal complement. (Contributed by NM, 8-Aug-2000.) (New usage is discouraged.)

Theoremoccon2 22710 Double contraposition for orthogonal complement. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)

Theoremoccon2i 22711 Double contraposition for orthogonal complement. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)

Theoremoc0 22712 The zero vector belongs to an orthogonal complement of a Hilbert subspace. (Contributed by NM, 11-Oct-1999.) (New usage is discouraged.)

Theoremocorth 22713 Members of a subset and its complement are orthogonal. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)

Theoremshocorth 22714 Members of a subspace and its complement are orthogonal. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)

Theoremococss 22715 Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)

Theoremshococss 22716 Inclusion in complement of complement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 10-Oct-1999.) (New usage is discouraged.)

Theoremshorth 22717 Members of orthogonal subspaces are orthogonal. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)

Theoremocin 22718 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 22719 Hilbert lattice contraposition law. (Contributed by Mario Carneiro, 18-May-2014.) (New usage is discouraged.)

Theoremocnel 22720 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 22721* 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 22722 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 22723 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 22724* 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 22725 Lemma for occl 22726. (Contributed by NM, 7-Aug-2000.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)

Theoremoccl 22726 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 22727 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 22728 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 22729 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 22730* Define subspace sum in . See shsval 22734, shsval2i 22809, and shsval3i 22810 for its value. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)

Definitiondf-span 22731* Define the linear span of a subset of Hilbert space. Definition of span in [Schechter] p. 276. See spanval 22755 for its value. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)

Definitiondf-chj 22732* Define Hilbert lattice join. See chjval 22774 for its value and chjcl 22779 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 22777. (Contributed by NM, 1-Nov-2000.) (New usage is discouraged.)

Definitiondf-chsup 22733 Define the supremum of a set of Hilbert lattice elements. See chsupval2 22832 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 22761. (Contributed by NM, 9-Dec-2003.) (New usage is discouraged.)

Theoremshsval 22734 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 22735 The subspace sum is a subset of Hilbert space. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremshsel 22736* 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 22737* Membership in the subspace sum of two Hilbert subspaces, using vector subtraction. (Contributed by NM, 20-Jan-2007.) (New usage is discouraged.)

Theoremshseli 22738* Membership in subspace sum. (Contributed by NM, 4-May-2000.) (New usage is discouraged.)

Theoremshscli 22739 Closure of subspace sum. (Contributed by NM, 15-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremshscl 22740 Closure of subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)

Theoremshscom 22741 Commutative law for subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)

Theoremshsva 22742 Vector sum belongs to subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)

Theoremshsel1 22743 A subspace sum contains a member of one of its subspaces. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)

Theoremshsel2 22744 A subspace sum contains a member of one of its subspaces. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)

Theoremshsvs 22745 Vector subtraction belongs to subspace sum. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)

Theoremshsub1 22746 Subspace sum is an upper bound of its arguments. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.)

Theoremshsub2 22747 Subspace sum is an upper bound of its arguments. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)

Theoremchoc0 22748 The orthocomplement of the zero subspace is the unit subspace. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Theoremchoc1 22749 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 22750 Orthogonal complement of the empty set. (Contributed by NM, 31-Oct-2000.) (New usage is discouraged.)

Theoremshintcli 22751 Closure of intersection of a non-empty subset of . (Contributed by NM, 14-Oct-1999.) (New usage is discouraged.)

Theoremshintcl 22752 The intersection of a non-empty set of subspaces is a subspace. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)

Theoremchintcli 22753 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 22754 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 22755* 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 22756 Value of supremum of set of subsets of Hilbert space. For an alternate version of the value, see hsupval2 22831. (Contributed by NM, 9-Dec-2003.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremchsupval 22757 The value of the supremum of a set of closed subspaces of Hilbert space. For an alternate version of the value, see chsupval2 22832. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)

Theoremspancl 22758 The span of a subset of Hilbert space is a subspace. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)

Theoremelspancl 22759 A member of a span is a vector. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)

Theoremshsupcl 22760 Closure of the subspace supremum of set of subsets of Hilbert space. (Contributed by NM, 26-Nov-2004.) (New usage is discouraged.)

Theoremhsupcl 22761 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 22762 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 22763 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 22764 Subset relation for supremum of subset of . (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)

Theoremhsupunss 22765 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 22766 The union of a set of closed subspaces is smaller than its supremum. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)

Theoremspanss2 22767 A subset of Hilbert space is included in its span. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)

Theoremshsupunss 22768 The union of a set of subspaces is smaller than its supremum. (Contributed by NM, 26-Nov-2004.) (New usage is discouraged.)

Theoremspanid 22769 A subspace of Hilbert space is its own span. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)

Theoremspanss 22770 Ordering relationship for the spans of subsets of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)

Theoremspanssoc 22771 The span of a subset of Hilbert space is less than or equal to its closure (double orthogonal complement). (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)

Theoremsshjval 22772 Value of join for subsets of Hilbert space. (Contributed by NM, 1-Nov-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremshjval 22773 Value of join in . (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)

Theoremchjval 22774 Value of join in . (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)

Theoremchjvali 22775 Value of join in . (Contributed by NM, 9-Aug-2000.) (New usage is discouraged.)

Theoremsshjval3 22776 Value of join for subsets of Hilbert space in terms of supremum: the join is the supremum of its two arguments. Based on the definition of join in [Beran] p. 3. For later convenience we prove a general version that works for any subset of Hilbert space, not just the elements of the lattice . (Contributed by NM, 2-Mar-2004.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremsshjcl 22777 Closure of join for subsets of Hilbert space. (Contributed by NM, 1-Nov-2000.) (New usage is discouraged.)

Theoremshjcl 22778 Closure of join in . (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)

Theoremchjcl 22779 Closure of join in . (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)

Theoremshjcom 22780 Commutative law for Hilbert lattice join of subspaces. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)

Theoremshless 22781 Subset implies subset of subspace sum. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremshlej1 22782 Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 22-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremshlej2 22783 Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)

Theoremshincli 22784 Closure of intersection of two subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Theoremshscomi 22785 Commutative law for subspace sum. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)

Theoremshsvai 22786 Vector sum belongs to subspace sum. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)

Theoremshsel1i 22787 A subspace sum contains a member of one of its subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Theoremshsel2i 22788 A subspace sum contains a member of one of its subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Theoremshsvsi 22789 Vector subtraction belongs to subspace sum. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Theoremshunssi 22790 Union is smaller than subspace sum. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)

Theoremshunssji 22791 Union is smaller than Hilbert lattice join. (Contributed by NM, 11-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremshsleji 22792 Subspace sum is smaller than Hilbert lattice join. Remark in [Kalmbach] p. 65. (Contributed by NM, 19-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremshjcomi 22793 Commutative law for join in . (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Theoremshsub1i 22794 Subspace sum is an upper bound of its arguments. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Theoremshsub2i 22795 Subspace sum is an upper bound of its arguments. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)

Theoremshub1i 22796 Hilbert lattice join is an upper bound of two subspaces. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Theoremshjcli 22797 Closure of join. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Theoremshjshcli 22798 closure of join. (Contributed by NM, 5-May-2000.) (New usage is discouraged.)

Theoremshlessi 22799 Subset implies subset of subspace sum. (Contributed by NM, 18-Nov-2000.) (New usage is discouraged.)

Theoremshlej1i 22800 Add disjunct to both sides of Hilbert subspace ordering. (Contributed by NM, 19-Oct-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

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