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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ssipeq 20001 | The inner product on a subspace equals the inner product on the parent space. (Contributed by AV, 19-Oct-2021.) |
↾s | ||
Theorem | phssipval 20002 | The inner product on a subspace in terms of the inner product on the parent space. (Contributed by NM, 28-Jan-2008.) (Revised by AV, 19-Oct-2021.) |
↾s | ||
Theorem | phssip 20003 | The inner product (as a function) on a subspace is a restriction of the inner product (as a function) on the parent space. (Contributed by NM, 28-Jan-2008.) (Revised by AV, 19-Oct-2021.) |
↾s | ||
Syntax | cocv 20004 | Extend class notation with orthocomplement of a subspace. |
Syntax | ccss 20005 | Extend class notation with set of closed subspaces. |
Syntax | cthl 20006 | Extend class notation with the Hilbert lattice. |
toHL | ||
Definition | df-ocv 20007* | Define orthocomplement of a subspace. (Contributed by NM, 7-Oct-2011.) |
Scalar | ||
Definition | df-css 20008* | Define set of closed subspaces. (Contributed by NM, 7-Oct-2011.) |
Definition | df-thl 20009 | Define the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) |
toHL toInc sSet | ||
Theorem | ocvfval 20010* | The orthocomplement operation. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.) |
Scalar | ||
Theorem | ocvval 20011* | Value of the orthocomplement of a subset (normally a subspace) of a pre-Hilbert space. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.) |
Scalar | ||
Theorem | elocv 20012* | Elementhood in the orthocomplement of a subset (normally a subspace) of a pre-Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Scalar | ||
Theorem | ocvi 20013 | Property of a member of the orthocomplement of a subset. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Scalar | ||
Theorem | ocvss 20014 | The orthocomplement of a subset is a subset of the base. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Theorem | ocvocv 20015 | A set is contained in its double orthocomplement. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Theorem | ocvlss 20016 | The orthocomplement of a subset is a linear subspace of the pre-Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Theorem | ocv2ss 20017 | Orthocomplements reverse subset inclusion. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Theorem | ocvin 20018 | An orthocomplement has trivial intersection with the original subspace. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Theorem | ocvsscon 20019 | Two ways to say that and are orthogonal subspaces. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Theorem | ocvlsp 20020 | The orthocomplement of a linear span. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Theorem | ocv0 20021 | The orthocomplement of the empty set. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Theorem | ocvz 20022 | The orthocomplement of the zero subspace. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Theorem | ocv1 20023 | The orthocomplement of the base set. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Theorem | unocv 20024 | The orthocomplement of a union. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Theorem | iunocv 20025* | The orthocomplement of an indexed union. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Theorem | cssval 20026* | The set of closed subspaces of a pre-Hilbert space. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.) |
Theorem | iscss 20027 | The predicate "is a closed subspace" (of a pre-Hilbert space). (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.) |
Theorem | cssi 20028 | Property of a closed subspace (of a pre-Hilbert space). (Contributed by Mario Carneiro, 13-Oct-2015.) |
Theorem | cssss 20029 | A closed subspace is a subset of the base. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Theorem | iscss2 20030 | 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.) |
Theorem | ocvcss 20031 | The orthocomplement of any set is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Theorem | cssincl 20032 | The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Theorem | css0 20033 | The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Theorem | css1 20034 | The whole space is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Theorem | csslss 20035 | A closed subspace of a pre-Hilbert space is a linear subspace. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Theorem | lsmcss 20036 | 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.) |
Theorem | cssmre 20037 | 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 16249: 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 16315. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Moore | ||
Theorem | mrccss 20038 | The Moore closure corresponding to the system of closed subspaces is the double orthocomplement operation. (Contributed by Mario Carneiro, 13-Oct-2015.) |
mrCls | ||
Theorem | thlval 20039 | Value of the Hilbert lattice. (Contributed by Mario Carneiro, 25-Oct-2015.) |
toHL toInc sSet | ||
Theorem | thlbas 20040 | Base set of the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) |
toHL | ||
Theorem | thlle 20041 | Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) |
toHL toInc | ||
Theorem | thlleval 20042 | Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) |
toHL | ||
Theorem | thloc 20043 | Orthocomplement on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) |
toHL | ||
Syntax | cpj 20044 | Extend class notation with orthogonal projection function. |
Syntax | chs 20045 | Extend class notation with class of all Hilbert spaces. |
Syntax | cobs 20046 | Extend class notation with the set of orthonormal bases. |
OBasis | ||
Definition | df-pj 20047* | Define orthogonal projection onto a subspace. This is just a wrapping of df-pj1 18052, but we restrict the domain of this function to only total projection functions. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Definition | df-hil 20048 | 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.) |
Definition | df-obs 20049* | 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 | ||
Theorem | pjfval 20050* | The value of the projection function. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Theorem | pjdm 20051 | 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.) |
Theorem | pjpm 20052 | The projection map is a partial function from subspaces of the pre-Hilbert space to total operators. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Theorem | pjfval2 20053* | Value of the projection map with implicit domain. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Theorem | pjval 20054 | Value of the projection map. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Theorem | pjdm2 20055 | 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.) |
Theorem | pjff 20056 | A projection is a linear operator. (Contributed by Mario Carneiro, 16-Oct-2015.) |
LMHom | ||
Theorem | pjf 20057 | A projection is a function on the base set. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Theorem | pjf2 20058 | A projection is a function from the base set to the subspace. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Theorem | pjfo 20059 | A projection is a surjection onto the subspace. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Theorem | pjcss 20060 | A projection subspace is an (algebraically) closed subspace. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Theorem | ocvpj 20061 | The orthocomplement of a projection subspace is a projection subspace. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Theorem | ishil 20062 | 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.) |
Theorem | ishil2 20063* | The predicate "is a Hilbert space" (over a *-division ring). (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 22-Jun-2014.) |
Theorem | isobs 20064* | The predicate "is an orthonormal basis" (over a pre-Hilbert space). (Contributed by Mario Carneiro, 23-Oct-2015.) |
Scalar OBasis | ||
Theorem | obsip 20065 | The inner product of two elements of an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Scalar OBasis | ||
Theorem | obsipid 20066 | A basis element has unit length. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Scalar OBasis | ||
Theorem | obsrcl 20067 | Reverse closure for an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.) |
OBasis | ||
Theorem | obsss 20068 | An orthonormal basis is a subset of the base set. (Contributed by Mario Carneiro, 23-Oct-2015.) |
OBasis | ||
Theorem | obsne0 20069 | A basis element is nonzero. (Contributed by Mario Carneiro, 23-Oct-2015.) |
OBasis | ||
Theorem | obsocv 20070 | An orthonormal basis has trivial orthocomplement. (Contributed by Mario Carneiro, 23-Oct-2015.) |
OBasis | ||
Theorem | obs2ocv 20071 | The double orthocomplement (closure) of an orthonormal basis is the whole space. (Contributed by Mario Carneiro, 23-Oct-2015.) |
OBasis | ||
Theorem | obselocv 20072 | 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 | ||
Theorem | obs2ss 20073 | A basis has no proper subsets that are also bases. (Contributed by Mario Carneiro, 23-Oct-2015.) |
OBasis OBasis | ||
Theorem | obslbs 20074 | 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 | ||
According to Wikipedia ("Linear algebra", 03-Mar-2019, https://en.wikipedia.org/wiki/Linear_algebra) "Linear algebra is the branch of mathematics concerning linear equations [...], linear functions [...] and their representations through matrices and vector spaces." Or according to the Merriam-Webster dictionary ("linear algebra", 12-Mar-2019, https://www.merriam-webster.com/dictionary/linear%20algebra) "Definition of linear algebra: a branch of mathematics that is concerned with mathematical structures closed under the operations of addition and scalar multiplication and that includes the theory of systems of linear equations, matrices, determinants, vector spaces, and linear transformations." However, dealing with modules (over rings) instead of vector spaces (over fields) allows for a more general approach. Therefore, "vectors" are regarded as members (elements of the base set) of a (free) module over a ring (see df-frlm 20091) in the following. By this, the number of entries in a vector is determined by the size of the index set of the direct sum building the free module the vector is belonging to. Since every vector space is isomorphic to a free module (see lvecisfrlm 20182), the theorems stated for free modules are also valid for vector spaces. Until not explicitly stated, the underlying ring needs not to be commutative (see df-cring 18550), but the existence of a multiplicative neutral element is always presumed (the ring is a unital ring, see also df-ring 18549). In this sense, linear equations, matrices and determinants are usually regarded as "over a ring" in this part. | ||
According to Wikipedia ("Direct sum of modules", 28-Mar-2019,
https://en.wikipedia.org/wiki/Direct_sum_of_modules) "Let R be a ring, and
{ Mi: i ∈ I } a family of left R-modules indexed by the set I.
The direct sum of {Mi} is then defined to be the set of all
sequences (αi) where αi ∈ Mi
and αi = 0 for cofinitely many indices i. (The direct product
is analogous but the indices do not need to cofinitely vanish.)". In this
definition, "cofinitely many" means "almost all" or "for all but finitely
many". Furthemore, "This set inherits the module structure via componentwise
addition and scalar multiplication. Explicitly, two such sequences α and
β can be added by writing (α + β)i =
αi + βi for all i (note that this is again
zero for all but finitely many indices), and such a sequence can be multiplied
with an element r from R by defining r(α)i =
(rα)i for all i.".
| ||
Syntax | cdsmm 20075 | Class of module direct sum generator. |
m | ||
Definition | df-dsmm 20076* | The direct sum of a family of Abelian groups or left modules is the induced group structure on finite linear combinations of elements, here represented as functions with finite support. (Contributed by Stefan O'Rear, 7-Jan-2015.) |
m s ↾s | ||
Theorem | reldmdsmm 20077 | The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) |
m | ||
Theorem | dsmmval 20078* | Value of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.) |
s m s ↾s | ||
Theorem | dsmmbase 20079* | Base set of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.) |
s m | ||
Theorem | dsmmval2 20080 | Self-referential definition of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
m m s ↾s | ||
Theorem | dsmmbas2 20081* | Base set of the direct sum module using the fndmin 6324 abbreviation. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
s m | ||
Theorem | dsmmfi 20082 | For finite products, the direct sum is just the module product. See also the observation in [Lang] p. 129. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
m s | ||
Theorem | dsmmelbas 20083* | Membership in the finitely supported hull of a structure product in terms of the index set. (Contributed by Stefan O'Rear, 11-Jan-2015.) |
s m | ||
Theorem | dsmm0cl 20084 | The all-zero vector is contained in the finite hull, since its support is empty and therefore finite. This theorem along with the next one effectively proves that the finite hull is a "submonoid", although that does not exist as a defined concept yet. (Contributed by Stefan O'Rear, 11-Jan-2015.) |
s m | ||
Theorem | dsmmacl 20085 | The finite hull is closed under addition. (Contributed by Stefan O'Rear, 11-Jan-2015.) |
s m | ||
Theorem | prdsinvgd2 20086 | Negation of a single coordinate in a structure product. (Contributed by Stefan O'Rear, 11-Jan-2015.) |
s | ||
Theorem | dsmmsubg 20087 | The finite hull of a product of groups is additionally closed under negation and thus is a subgroup of the product. (Contributed by Stefan O'Rear, 11-Jan-2015.) |
s m SubGrp | ||
Theorem | dsmmlss 20088* | The finite hull of a product of modules is additionally closed under scalar multiplication and thus is a linear subspace of the product. (Contributed by Stefan O'Rear, 11-Jan-2015.) |
Scalar s m | ||
Theorem | dsmmlmod 20089* | The direct sum of a family of modules is a module. See also the remark in [Lang] p. 128. (Contributed by Stefan O'Rear, 11-Jan-2015.) |
Scalar m | ||
According to Wikipedia ("Free module", 03-Mar-2019, https://en.wikipedia.org/wiki/Free_module) "In mathematics, a free module is a module that has a basis - that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules.". The same definition is used in [Lang] p. 135: "By a free module we shall mean a module which admits a basis, or the zero module.". In the following, however, a free module is defined as direct sum of a family consisting of the same ring regarded as a (left) module over itself, see df-frlm 20091. Since a module has a basis if and only if it is isomorphic to a free module as defined by df-frlm 20091 (see lmisfree 20181), the two definitions are essentially equivalent. The free modules as defined by df-frlm 20091 are also taken for the motivation of free modules by [Lang] p. 135. | ||
Syntax | cfrlm 20090 | Class of free module generator. |
freeLMod | ||
Definition | df-frlm 20091* | The -dimensional free module over a ring is the product of -many copies of the ring with componentwise addition and multiplication. If is infinite, the allowed vectors are restricted to those with finitely many nonzero coordinates; this ensures that the resulting module is actually spanned by its unit vectors. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
freeLMod m ringLMod | ||
Theorem | frlmval 20092 | Value of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
freeLMod m ringLMod | ||
Theorem | frlmlmod 20093 | The free module is a module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
freeLMod | ||
Theorem | frlmpws 20094 | The free module as a restriction of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
freeLMod ringLMod s ↾s | ||
Theorem | frlmlss 20095 | The base set of the free module is a subspace of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
freeLMod ringLMod s | ||
Theorem | frlmpwsfi 20096 | The finite free module is a power of the ring module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
freeLMod ringLMod s | ||
Theorem | frlmsca 20097 | The ring of scalars of a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
freeLMod Scalar | ||
Theorem | frlm0 20098 | Zero in a free module (ring constraint is stronger than necessary, but allows use of frlmlss 20095). (Contributed by Stefan O'Rear, 4-Feb-2015.) |
freeLMod | ||
Theorem | frlmbas 20099* | Base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 23-Jun-2019.) |
freeLMod finSupp | ||
Theorem | frlmelbas 20100 | Membership in the base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 23-Jun-2019.) |
freeLMod finSupp |
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