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Theorem List for Metamath Proof Explorer - 21901-22000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremshjshsi 21901 Hilbert lattice join equals the double orthocomplement of subspace sum. (Contributed by NM, 27-Nov-2004.) (New usage is discouraged.)

Theoremshjshseli 21902 A closed subspace sum equals Hilbert lattice join. Part of Lemma 31.1.5 of [MaedaMaeda] p. 136. (Contributed by NM, 30-Nov-2004.) (New usage is discouraged.)

Theoremchne0 21903* A nonzero closed subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)

Theoremchocin 21904 Intersection of a closed subspace and its orthocomplement. Part of Proposition 1 of [Kalmbach] p. 65. (Contributed by NM, 13-Jun-2006.) (New usage is discouraged.)

Theoremchssoc 21905 A closed subspace less than its orthocomplement is zero. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)

Theoremchj0 21906 Join with Hilbert lattice zero. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)

Theoremchslej 21907 Subspace sum is smaller than subspace join. Remark in [Kalmbach] p. 65. (Contributed by NM, 12-Jul-2004.) (New usage is discouraged.)

Theoremchincl 21908 Closure of Hilbert lattice intersection. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)

Theoremchsscon3 21909 Hilbert lattice contraposition law. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremchsscon1 21910 Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchsscon2 21911 Hilbert lattice contraposition law. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchpsscon3 21912 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremchpsscon1 21913 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremchpsscon2 21914 Hilbert lattice contraposition law for strict ordering. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremchjcom 21915 Commutative law for Hilbert lattice join. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremchub1 21916 Hilbert lattice join is greater than or equal to its first argument. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremchub2 21917 Hilbert lattice join is greater than or equal to its second argument. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremchlub 21918 Hilbert lattice join is the least upper bound of two elements. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Theoremchlej1 21919 Add join to both sides of Hilbert lattice ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)

Theoremchlej2 21920 Add join to both sides of Hilbert lattice ordering. (Contributed by NM, 22-Jun-2004.) (New usage is discouraged.)

Theoremchlejb1 21921 Hilbert lattice ordering in terms of join. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.)

Theoremchlejb2 21922 Hilbert lattice ordering in terms of join. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)

Theoremchnle 21923 Equivalent expressions for "not less than" in the Hilbert lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)

Theoremchjo 21924 The join of a closed subspace and its orthocomplement is all of Hilbert space. (Contributed by NM, 31-Oct-2005.) (New usage is discouraged.)

Theoremchabs1 21925 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)

Theoremchabs2 21926 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)

Theoremchabs1i 21927 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)

Theoremchabs2i 21928 Hilbert lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 16-Jun-2004.) (New usage is discouraged.)

Theoremchjidm 21929 Idempotent law for Hilbert lattice join. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)

Theoremchjidmi 21930 Idempotent law for Hilbert lattice join. (Contributed by NM, 15-Jun-2004.) (New usage is discouraged.)

Theoremchj12i 21931 A rearrangement of Hilbert lattice join. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremchj4i 21932 Rearrangement of the join of 4 Hilbert lattice elements. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremchjjdiri 21933 Hilbert lattice join distributes over itself. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)

Theoremchdmm1 21934 DeMorgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmm2 21935 DeMorgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmm3 21936 DeMorgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmm4 21937 DeMorgan's law for meet in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmj1 21938 DeMorgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmj2 21939 DeMorgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmj3 21940 DeMorgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchdmj4 21941 DeMorgan's law for join in a Hilbert lattice. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.)

Theoremchjass 21942 Associative law for Hilbert lattice join. From definition of lattice in [Kalmbach] p. 14. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)

Theoremchj12 21943 A rearrangement of Hilbert lattice join. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)

Theoremchj4 21944 Rearrangement of the join of 4 Hilbert lattice elements. (Contributed by NM, 15-Jun-2006.) (New usage is discouraged.)

Theoremledii 21945 An ortholattice is distributive in one ordering direction. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.)

Theoremlediri 21946 An ortholattice is distributive in one ordering direction. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Theoremlejdii 21947 An ortholattice is distributive in one ordering direction (join version). (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Theoremlejdiri 21948 An ortholattice is distributive in one ordering direction (join version). (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Theoremledi 21949 An ortholattice is distributive in one ordering direction. (Contributed by NM, 14-Jun-2006.) (New usage is discouraged.)

15.9.24  Span (cont.) and one-dimensional subspaces

Theoremspansn0 21950 The span of the singleton of the zero vector is the zero subspace. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.)

Theoremspan0 21951 The span of the empty set is the zero subspace. Remark 11.6.e of [Schechter] p. 276. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)

Theoremelspani 21952* Membership in the span of a subset of Hilbert space. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)

Theoremspanuni 21953 The span of a union is the subspace sum of spans. (Contributed by NM, 2-Jun-2004.) (New usage is discouraged.)

Theoremspanun 21954 The span of a union is the subspace sum of spans. (Contributed by NM, 9-Jun-2006.) (New usage is discouraged.)

Theoremsshhococi 21955 The join of two Hilbert space subsets (not necessarily closed subspaces) equals the join of their closures (double orthocomplements). (Contributed by NM, 1-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremhne0 21956 Hilbert space has a nonzero vector iff it is not trivial. (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.)

Theoremchsup0 21957 The supremum of the empty set. (Contributed by NM, 13-Aug-2002.) (New usage is discouraged.)

Theoremh1deoi 21958 Membership in orthocomplement of 1-dimensional subspace. (Contributed by NM, 7-Jul-2001.) (New usage is discouraged.)

Theoremh1dei 21959* Membership in 1-dimensional subspace. (Contributed by NM, 7-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremh1did 21960 A generating vector belongs to the 1-dimensional subspace it generates. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)

Theoremh1dn0 21961 A nonzero vector generates a (nonzero) 1-dimensional subspace. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)

Theoremh1de2i 21962 Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 17-Jul-2001.) (New usage is discouraged.)

Theoremh1de2bi 21963 Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 19-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremh1de2ctlem 21964* Lemma for h1de2ci 21965. (Contributed by NM, 19-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremh1de2ci 21965* Membership in 1-dimensional subspace. All members are collinear with the generating vector. (Contributed by NM, 21-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremspansni 21966 The span of a singleton in Hilbert space equals its closure. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)

Theoremelspansni 21967* Membership in the span of a singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)

Theoremspansn 21968 The span of a singleton in Hilbert space equals its closure. (Contributed by NM, 4-Jun-2004.) (New usage is discouraged.)

Theoremspansnch 21969 The span of a Hilbert space singleton belongs to the Hilbert lattice. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)

Theoremspansnsh 21970 The span of a Hilbert space singleton is a subspace. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)

Theoremspansnchi 21971 The span of a singleton in Hilbert space is a closed subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)

Theoremspansnid 21972 A vector belongs to the span of its singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)

Theoremspansnmul 21973 A scalar product with a vector belongs to the span of its singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)

Theoremelspansncl 21974 A member of a span of a singleton is a vector. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)

Theoremelspansn 21975* Membership in the span of a singleton. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)

Theoremelspansn2 21976 Membership in the span of a singleton. All members are collinear with the generating vector. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)

Theoremspansncol 21977 The singletons of collinear vectors have the same span. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)

Theoremspansneleqi 21978 Membership relation implied by equality of spans. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)

Theoremspansneleq 21979 Membership relation that implies equality of spans. (Contributed by NM, 6-Jun-2004.) (New usage is discouraged.)

Theoremspansnss 21980 The span of the singleton of an element of a subspace is included in the subspace. (Contributed by NM, 5-Jun-2004.) (New usage is discouraged.)

Theoremelspansn3 21981 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (Contributed by NM, 16-Dec-2004.) (New usage is discouraged.)

Theoremelspansn4 21982 A span membership condition implying two vectors belong to the same subspace. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)

Theoremelspansn5 21983 A vector belonging to both a subspace and the span of the singleton of a vector not in it must be zero. (Contributed by NM, 17-Dec-2004.) (New usage is discouraged.)

Theoremspansnss2 21984 The span of the singleton of an element of a subspace is included in the subspace. (Contributed by NM, 16-Dec-2004.) (New usage is discouraged.)

Theoremnormcan 21985 Cancellation-type law that "extracts" a vector from its inner product with a proportional vector . (Contributed by NM, 18-Mar-2006.) (New usage is discouraged.)

Theorempjspansn 21986 A projection on the span of a singleton. (The proof ws shortened by Mario Carneiro, 15-Dec-2013.) (Contributed by NM, 28-May-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Theoremspansnpji 21987 A subset of Hilbert space is orthogonal to the span of the singleton of a projection onto its orthocomplement. (Contributed by NM, 4-Jun-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)

Theoremspanunsni 21988 The span of the union of a closed subspace with a singleton equals the span of its union with an orthogonal singleton. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)

Theoremspanpr 21989 The span of a pair of vectors. (Contributed by NM, 9-Jun-2006.) (New usage is discouraged.)

Theoremh1datomi 21990 A 1-dimensional subspace is an atom. (Contributed by NM, 20-Jul-2001.) (New usage is discouraged.)

Theoremh1datom 21991 A 1-dimensional subspace is an atom. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.)

15.9.25  Operator sum, difference, and scalar multiplication

Note on operators. Unlike some authors, we use the term "operator" to mean any function from to . This is the definition of operator in [Hughes] p. 14, the definition of operator in [AkhiezerGlazman] p. 30, and the definition of operator in [Goldberg] p. 10. For Reed and Simon, an operator is linear (definition of operator in [ReedSimon] p. 2). For Halmos, an operator is bounded and linear (definition of operator in [Halmos] p. 35). For Kalmbach and Beran, an operator is continuous and linear (definition of operator in [Kalmbach] p. 353; definition of operator in [Beran] p. 99). Note that "bounded and linear" and "continuous and linear" are equivalent by lncnbd 22448.

Definitiondf-hosum 21992* Define the sum of two Hilbert space operators. Definition of [Beran] p. 111. (Contributed by NM, 9-Nov-2000.) (New usage is discouraged.)

Definitiondf-homul 21993* Define the scalar product with a Hilbert space operator. Definition of [Beran] p. 111. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)

Definitiondf-hodif 21994* Define the difference of two Hilbert space operators. Definition of [Beran] p. 111. (Contributed by NM, 9-Nov-2000.) (New usage is discouraged.)

Definitiondf-hfsum 21995* Define the sum of two Hilbert space functionals. Definition of [Beran] p. 111. Note that unlike some authors, we define a functional as any function from to , not just linear (or bounded linear) ones. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)

Definitiondf-hfmul 21996* Define the scalar product with a Hilbert space functional. Definition of [Beran] p. 111. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)

Theoremhosmval 21997* Value of the sum of two Hilbert space operators. (Contributed by NM, 9-Nov-2000.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)

Theoremhommval 21998* Value of the scalar product with a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)

Theoremhodmval 21999* Value of the difference of two Hilbert space operators. (Contributed by NM, 9-Nov-2000.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)

Theoremhfsmval 22000* Value of the sum of two Hilbert space functionals. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)

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