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Theorem List for Metamath Proof Explorer - 16901-17000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremiscss2 16901 It is sufficient to prove that the double orthocomplement is a subset of the target set to show that the set is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)

Theoremocvcss 16902 The orthocomplement of any set is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)

Theoremcssincl 16903 The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)

Theoremcss0 16904 The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)

Theoremcss1 16905 The whole space is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)

Theoremcsslss 16906 A closed subspace of a pre-Hilbert space is a linear subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)

Theoremlsmcss 16907 A subset of a pre-Hilbert space whose double orthocomplement has a projection decomposition is a closed subspace. This is the core of the proof that a topologically closed subspace is algebraically closed in a Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.)

Theoremcssmre 16908 The closed subspaces of a pre-Hilbert space are a Moore system. Unlike many of our other examples of closure systems, this one is not usually an algebraic closure system df-acs 13802: consider the Hilbert space of sequences with convergent sum; the subspace of all sequences with finite support is the classic example of a non-closed subspace, but for every finite set of sequences of finite support, there is a finite-dimensional (and hence closed) subspace containing all of the sequences, so if closed subspaces were an algebraic closure system this would violate acsfiel 13867. (Contributed by Mario Carneiro, 13-Oct-2015.)
Moore

Theoremmrccss 16909 The Moore closure corresponding to the system of closed subspaces is the double orthocomplement operation. (Contributed by Mario Carneiro, 13-Oct-2015.)
mrCls

Theoremthlval 16910 Value of the Hilbert lattice. (Contributed by Mario Carneiro, 25-Oct-2015.)
toHL              toInc              sSet

Theoremthlbas 16911 Base set of the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
toHL

Theoremthlle 16912 Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
toHL              toInc

Theoremthlleval 16913 Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
toHL

Theoremthloc 16914 Orthocomplement on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
toHL

10.12.3  Orthogonal projection and orthonormal bases

Syntaxcpj 16915 Extend class notation with orthogonal projection function.

Syntaxchs 16916 Extend class notation with class of all Hilbert spaces.

Syntaxcobs 16917 Extend class notation with the set of orthonormal bases.
OBasis

Definitiondf-pj 16918* Define orthogonal projection onto a subspace. This is just a wrapping of df-pj1 15259, but we restrict the domain of this function to only total projection functions. (Contributed by Mario Carneiro, 16-Oct-2015.)

Definitiondf-hil 16919 Define class of all Hilbert spaces. Based on Proposition 4.5, p. 176, Gudrun Kalmbach, Quantum Measures and Spaces, Kluwer, Dordrecht, 1998. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 16-Oct-2015.)

Definitiondf-obs 16920* Define the set of all orthonormal bases for a pre-Hilbert space. An orthonormal basis is a set of mutually orthogonal vectors with norm 1 and such that the linear span is dense in the whole space. (As this is an "algebraic" definition, before we have topology available, we express this denseness by saying that the double orthocomplement is the whole space, or equivalently, the single orthocomplement is trivial.) (Contributed by Mario Carneiro, 23-Oct-2015.)
OBasis Scalar Scalar

Theorempjfval 16921* The value of the projection function. (Contributed by Mario Carneiro, 16-Oct-2015.)

Theorempjdm 16922 A subspace is in the domain of the projection function iff the subspace admits a projection decomposition of the whole space. (Contributed by Mario Carneiro, 16-Oct-2015.)

Theorempjpm 16923 The projection map is a partial function from subspaces of the pre-Hilbert space to total operators. (Contributed by Mario Carneiro, 16-Oct-2015.)

Theorempjfval2 16924* Value of the projection map with implicit domain. (Contributed by Mario Carneiro, 16-Oct-2015.)

Theorempjval 16925 Value of the projection map. (Contributed by Mario Carneiro, 16-Oct-2015.)

Theorempjdm2 16926 A subspace is in the domain of the projection function iff the subspace admits a projection decomposition of the whole space. (Contributed by Mario Carneiro, 16-Oct-2015.)

Theorempjff 16927 A projection is a linear operator. (Contributed by Mario Carneiro, 16-Oct-2015.)
LMHom

Theorempjf 16928 A projection is a function on the base set. (Contributed by Mario Carneiro, 16-Oct-2015.)

Theorempjf2 16929 A projection is a function from the base set to the subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)

Theorempjfo 16930 A projection is a surjection onto the subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)

Theorempjcss 16931 A projection subspace is an (algebraically) closed subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)

Theoremocvpj 16932 The orthocomplement of a projection subspace is a projection subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)

Theoremishil 16933 The predicate "is a Hilbert space" (over a *-division ring). A Hilbert space is a pre-Hilbert space such that all closed subspaces have a projection decomposition. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 22-Jun-2014.)

Theoremishil2 16934* The predicate "is a Hilbert space" (over a *-division ring). (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 22-Jun-2014.)

Theoremisobs 16935* The predicate "is an orthonormal basis" (over a pre-Hilbert space). (Contributed by Mario Carneiro, 23-Oct-2015.)
Scalar                                   OBasis

Theoremobsip 16936 The inner product of two elements of an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)
Scalar                     OBasis

Theoremobsipid 16937 A basis element has unit length. (Contributed by Mario Carneiro, 23-Oct-2015.)
Scalar              OBasis

Theoremobsrcl 16938 Reverse closure for an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)
OBasis

Theoremobsss 16939 An orthonormal basis is a subset of the base set. (Contributed by Mario Carneiro, 23-Oct-2015.)
OBasis

Theoremobsne0 16940 A basis element is nonzero. (Contributed by Mario Carneiro, 23-Oct-2015.)
OBasis

Theoremobsocv 16941 An orthonormal basis has trivial orthocomplement. (Contributed by Mario Carneiro, 23-Oct-2015.)
OBasis

Theoremobs2ocv 16942 The double orthocomplement (closure) of an orthonormal basis is the whole space. (Contributed by Mario Carneiro, 23-Oct-2015.)
OBasis

Theoremobselocv 16943 A basis element is in the orthocomplement of a subset of the basis iff it is not in the subset. (Contributed by Mario Carneiro, 23-Oct-2015.)
OBasis

Theoremobs2ss 16944 A basis has no proper subsets that are also bases. (Contributed by Mario Carneiro, 23-Oct-2015.)
OBasis OBasis

Theoremobslbs 16945 An orthogonal basis is a linear basis iff the span of the basis elements is closed (which is usually not true). (Contributed by Mario Carneiro, 29-Oct-2015.)
LBasis                     OBasis

PART 11  BASIC TOPOLOGY

11.1  Topology

11.1.1  Topological spaces

Syntaxctop 16946 Extend class notation with the class of all topologies.

Syntaxctopon 16947 The class function of all topologies over a base set.
TopOn

SyntaxctpsOLD 16948 Extend class notation with the class of all topological spaces. (New usage is discouraged.)

Syntaxctps 16949 Extend class notation with the class of all topological spaces.

Syntaxctb 16950 Extend class notation with the class of all topological bases.

Definitiondf-top 16951* Define the (proper) class of all topologies. See istop2g 16957 for an alternate way to express finite intersection and istps5OLD 16977 for a standard definition in terms of both members of a topological space. (Contributed by NM, 3-Mar-2006.)

Definitiondf-topspOLD 16952* Define the class of all topological spaces, each of which is an ordered pair the second of which is a topology on the first. See istps5OLD 16977 for a standard way to express a topological space. (Contributed by NM, 8-Mar-2006.) (New usage is discouraged.)

Definitiondf-bases 16953* Define the class of all topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 17001). Note that "bases" is the plural of "basis." (Contributed by NM, 17-Jul-2006.)

Definitiondf-topon 16954* Define the set of topologies with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
TopOn

Definitiondf-topsp 16955 Define the class of all topological spaces (structures). (Contributed by Stefan O'Rear, 13-Aug-2015.)
TopOn

Theoremistopg 16956* Express the predicate " is a topology." Note: In the literature, a topology is often represented by a script letter T, which resembles the letter J. This confusion may have led to J being used by some authors - e.g. K. D. Joshi, Introduction to General Topology (1983), p. 114 - and it is convenient for us since we later use to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Theoremistop2g 16957* Express the predicate " is a topology," using "the intersection of the elements of any finite subcollection" instead of the intersection of any two elements. (Contributed by NM, 19-Jul-2006.)

Theoremuniopn 16958 The union of a subset of a topology is an open set. (Contributed by Stefan Allan, 27-Feb-2006.)

Theoremiunopn 16959* The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)

Theoreminopn 16960 The intersection of two open sets of a topology is also an open set. (Contributed by NM, 17-Jul-2006.)

Theoremfitop 16961 A topology is closed under finite intersections. (Contributed by Jeff Hankins, 7-Oct-2009.)

Theoremfiinopn 16962 The intersection of a non-empty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)

Theoremiinopn 16963* The intersection of a non-empty finite family of open sets is open. (Contributed by Mario Carneiro, 14-Sep-2014.)

Theoremunopn 16964 The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theorem0opn 16965 The empty set is an open subset of a topology. (Contributed by Stefan Allan, 27-Feb-2006.)

Theorem0ntop 16966 The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)

Theoremtopopn 16967 The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)

Theoremeltopss 16968 A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)

Theoremriinopn 16969* A finite indexed relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.)

Theoremrintopn 16970 A finite relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.)

TheoremeltopspOLD 16971 Construct a topological space from a topology and vice-versa. We say that is a topology on . (This could be proved more efficiently from istpsOLD 16973, but the proof here does not require the Axiom of Regularity.) (Contributed by NM, 8-Mar-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremtpsexOLD 16972 Existence implied by membership in a topological space. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremistpsOLD 16973 Express the predicate "is a topological space." (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremistps2OLD 16974 Express the predicate "is a topological space." (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremistps3OLD 16975* A standard textbook definition of a topological space. (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremistps4OLD 16976* A standard textbook definition of a topological space. (Contributed by NM, 19-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremistps5OLD 16977* A standard textbook definition of a topological space : a topology on is a collection of subsets of such that and are in and that is closed under union and finite intersection. Definition of topological space in [Munkres] p. 76. (Contributed by NM, 19-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremistopon 16978 Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtopontop 16979 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtoponuni 16980 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtoponmax 16981 The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtoponss 16982 A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.)
TopOn

Theoremtoponcom 16983 If is a topology on the base set of topology , then is a topology on the base of . (Contributed by Mario Carneiro, 22-Aug-2015.)
TopOn TopOn

Theoremtopontopi 16984 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtoponunii 16985 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtoptopon 16986 Alternative definition of in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremtopgele 16987 The topologies over the same set have the greatest element (the discrete topology) and the least element (the indiscrete topology). (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 16-Sep-2015.)
TopOn

Theoremtopsn 16988 The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 4001). (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
TopOn

Theoremistps 16989 Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.)
TopOn

Theoremistps2 16990 Express the predicate "is a topological space." (Contributed by NM, 20-Oct-2012.)

Theoremtpsuni 16991 The base set of a topological space. (Contributed by FL, 27-Jun-2014.)

Theoremtpstop 16992 The topology extractor on a topological space is a topology. (Contributed by FL, 27-Jun-2014.)

Theoremtpspropd 16993 A topological space depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)

Theoremtpsprop2d 16994 A topological space depends only on the base and topology components. (Contributed by Mario Carneiro, 13-Aug-2015.)
TopSet TopSet

Theoremtopontopn 16995 Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.)
TopSet       TopOn

Theoremtsettps 16996 If the topology component is already correctly truncated, then it forms a topological space (with the topology extractor function coming out the same as the component). (Contributed by Mario Carneiro, 13-Aug-2015.)
TopSet       TopOn

Theoremistpsi 16997 Properties that determine a topological space. (Contributed by NM, 20-Oct-2012.)

Theoremeltpsg 16998 Properties that determine a topological space from a construction (using no explicit indices). (Contributed by Mario Carneiro, 13-Aug-2015.)
TopSet        TopOn

Theoremeltpsi 16999 Properties that determine a topological space from a construction (using no explicit indices). (Contributed by NM, 20-Oct-2012.) (Revised by Mario Carneiro, 13-Aug-2015.)
TopSet

11.1.2  TopBases for topologies

Theoremisbasisg 17000* Express the predicate " is a basis for a topology." (Contributed by NM, 17-Jul-2006.)

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