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Theorem List for Metamath Proof Explorer - 20501-20600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiscss 20501 The predicate "is a closed subspace" (of a pre-Hilbert space). (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.)
= (ocv‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)       (𝑊𝑋 → (𝑆𝐶𝑆 = ( ‘( 𝑆))))
 
Theoremcssi 20502 Property of a closed subspace (of a pre-Hilbert space). (Contributed by Mario Carneiro, 13-Oct-2015.)
= (ocv‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)       (𝑆𝐶𝑆 = ( ‘( 𝑆)))
 
Theoremcssss 20503 A closed subspace is a subset of the base. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)       (𝑆𝐶𝑆𝑉)
 
Theoremiscss2 20504 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.)
𝑉 = (Base‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)    &    = (ocv‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝑆𝑉) → (𝑆𝐶 ↔ ( ‘( 𝑆)) ⊆ 𝑆))
 
Theoremocvcss 20505 The orthocomplement of any set is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)    &    = (ocv‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝑆𝑉) → ( 𝑆) ∈ 𝐶)
 
Theoremcssincl 20506 The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝐶 = (ClSubSp‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝐴𝐶𝐵𝐶) → (𝐴𝐵) ∈ 𝐶)
 
Theoremcss0 20507 The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝐶 = (ClSubSp‘𝑊)    &    0 = (0g𝑊)       (𝑊 ∈ PreHil → { 0 } ∈ 𝐶)
 
Theoremcss1 20508 The whole space is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)       (𝑊 ∈ PreHil → 𝑉𝐶)
 
Theoremcsslss 20509 A closed subspace of a pre-Hilbert space is a linear subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝐶 = (ClSubSp‘𝑊)    &   𝐿 = (LSubSp‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝑆𝐶) → 𝑆𝐿)
 
Theoremlsmcss 20510 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.)
𝐶 = (ClSubSp‘𝑊)    &   𝑉 = (Base‘𝑊)    &    = (ocv‘𝑊)    &    = (LSSum‘𝑊)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝑆𝑉)    &   (𝜑 → ( ‘( 𝑆)) ⊆ (𝑆 ( 𝑆)))       (𝜑𝑆𝐶)
 
Theoremcssmre 20511 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 16965: consider the Hilbert space of sequences ℕ⟶ℝ 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 17030. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)       (𝑊 ∈ PreHil → 𝐶 ∈ (Moore‘𝑉))
 
Theoremmrccss 20512 The Moore closure corresponding to the system of closed subspaces is the double orthocomplement operation. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &    = (ocv‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)    &   𝐹 = (mrCls‘𝐶)       ((𝑊 ∈ PreHil ∧ 𝑆𝑉) → (𝐹𝑆) = ( ‘( 𝑆)))
 
Theoremthlval 20513 Value of the Hilbert lattice. (Contributed by Mario Carneiro, 25-Oct-2015.)
𝐾 = (toHL‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)    &   𝐼 = (toInc‘𝐶)    &    = (ocv‘𝑊)       (𝑊𝑉𝐾 = (𝐼 sSet ⟨(oc‘ndx), ⟩))
 
Theoremthlbas 20514 Base set of the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
𝐾 = (toHL‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)       𝐶 = (Base‘𝐾)
 
Theoremthlle 20515 Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
𝐾 = (toHL‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)    &   𝐼 = (toInc‘𝐶)    &    = (le‘𝐼)        = (le‘𝐾)
 
Theoremthlleval 20516 Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
𝐾 = (toHL‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)    &    = (le‘𝐾)       ((𝑆𝐶𝑇𝐶) → (𝑆 𝑇𝑆𝑇))
 
Theoremthloc 20517 Orthocomplement on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
𝐾 = (toHL‘𝑊)    &    = (ocv‘𝑊)        = (oc‘𝐾)
 
10.9.3  Orthogonal projection and orthonormal bases
 
Syntaxcpj 20518 Extend class notation with orthogonal projection function.
class proj
 
Syntaxchil 20519 Extend class notation with class of all Hilbert spaces.
class Hil
 
Syntaxcobs 20520 Extend class notation with the set of orthonormal bases.
class OBasis
 
Definitiondf-pj 20521* Define orthogonal projection onto a subspace. This is just a wrapping of df-pj1 18882, but we restrict the domain of this function to only total projection functions. (Contributed by Mario Carneiro, 16-Oct-2015.)
proj = ( ∈ V ↦ ((𝑥 ∈ (LSubSp‘) ↦ (𝑥(proj1)((ocv‘)‘𝑥))) ∩ (V × ((Base‘) ↑m (Base‘)))))
 
Definitiondf-hil 20522 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.)
Hil = { ∈ PreHil ∣ dom (proj‘) = (ClSubSp‘)}
 
Definitiondf-obs 20523* 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 = ( ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Base‘) ∣ (∀𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ∧ ((ocv‘)‘𝑏) = {(0g)})})
 
Theorempjfval 20524* The value of the projection function. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐿 = (LSubSp‘𝑊)    &    = (ocv‘𝑊)    &   𝑃 = (proj1𝑊)    &   𝐾 = (proj‘𝑊)       𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉)))
 
Theorempjdm 20525 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.)
𝑉 = (Base‘𝑊)    &   𝐿 = (LSubSp‘𝑊)    &    = (ocv‘𝑊)    &   𝑃 = (proj1𝑊)    &   𝐾 = (proj‘𝑊)       (𝑇 ∈ dom 𝐾 ↔ (𝑇𝐿 ∧ (𝑇𝑃( 𝑇)):𝑉𝑉))
 
Theorempjpm 20526 The projection map is a partial function from subspaces of the pre-Hilbert space to total operators. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐿 = (LSubSp‘𝑊)    &   𝐾 = (proj‘𝑊)       𝐾 ∈ ((𝑉m 𝑉) ↑pm 𝐿)
 
Theorempjfval2 20527* Value of the projection map with implicit domain. (Contributed by Mario Carneiro, 16-Oct-2015.)
= (ocv‘𝑊)    &   𝑃 = (proj1𝑊)    &   𝐾 = (proj‘𝑊)       𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( 𝑥)))
 
Theorempjval 20528 Value of the projection map. (Contributed by Mario Carneiro, 16-Oct-2015.)
= (ocv‘𝑊)    &   𝑃 = (proj1𝑊)    &   𝐾 = (proj‘𝑊)       (𝑇 ∈ dom 𝐾 → (𝐾𝑇) = (𝑇𝑃( 𝑇)))
 
Theorempjdm2 20529 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.)
𝑉 = (Base‘𝑊)    &   𝐿 = (LSubSp‘𝑊)    &    = (ocv‘𝑊)    &    = (LSSum‘𝑊)    &   𝐾 = (proj‘𝑊)       (𝑊 ∈ PreHil → (𝑇 ∈ dom 𝐾 ↔ (𝑇𝐿 ∧ (𝑇 ( 𝑇)) = 𝑉)))
 
Theorempjff 20530 A projection is a linear operator. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐾 = (proj‘𝑊)       (𝑊 ∈ PreHil → 𝐾:dom 𝐾⟶(𝑊 LMHom 𝑊))
 
Theorempjf 20531 A projection is a function on the base set. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐾 = (proj‘𝑊)    &   𝑉 = (Base‘𝑊)       (𝑇 ∈ dom 𝐾 → (𝐾𝑇):𝑉𝑉)
 
Theorempjf2 20532 A projection is a function from the base set to the subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐾 = (proj‘𝑊)    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾𝑇):𝑉𝑇)
 
Theorempjfo 20533 A projection is a surjection onto the subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐾 = (proj‘𝑊)    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾𝑇):𝑉onto𝑇)
 
Theorempjcss 20534 A projection subspace is an (algebraically) closed subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐾 = (proj‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)       (𝑊 ∈ PreHil → dom 𝐾𝐶)
 
Theoremocvpj 20535 The orthocomplement of a projection subspace is a projection subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐾 = (proj‘𝑊)    &    = (ocv‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ( 𝑇) ∈ dom 𝐾)
 
Theoremishil 20536 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.)
𝐾 = (proj‘𝐻)    &   𝐶 = (ClSubSp‘𝐻)       (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶))
 
Theoremishil2 20537* The predicate "is a Hilbert space" (over a *-division ring). (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝑉 = (Base‘𝐻)    &    = (LSSum‘𝐻)    &    = (ocv‘𝐻)    &   𝐶 = (ClSubSp‘𝐻)       (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ ∀𝑠𝐶 (𝑠 ( 𝑠)) = 𝑉))
 
Theoremisobs 20538* The predicate "is an orthonormal basis" (over a pre-Hilbert space). (Contributed by Mario Carneiro, 23-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝐹 = (Scalar‘𝑊)    &    1 = (1r𝐹)    &    0 = (0g𝐹)    &    = (ocv‘𝑊)    &   𝑌 = (0g𝑊)       (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ 𝐵𝑉 ∧ (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝐵) = {𝑌})))
 
Theoremobsip 20539 The inner product of two elements of an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝐹 = (Scalar‘𝑊)    &    1 = (1r𝐹)    &    0 = (0g𝐹)       ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝑃𝐵𝑄𝐵) → (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 ))
 
Theoremobsipid 20540 A basis element has unit length. (Contributed by Mario Carneiro, 23-Oct-2015.)
, = (·𝑖𝑊)    &   𝐹 = (Scalar‘𝑊)    &    1 = (1r𝐹)       ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴𝐵) → (𝐴 , 𝐴) = 1 )
 
Theoremobsrcl 20541 Reverse closure for an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)
(𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil)
 
Theoremobsss 20542 An orthonormal basis is a subset of the base set. (Contributed by Mario Carneiro, 23-Oct-2015.)
𝑉 = (Base‘𝑊)       (𝐵 ∈ (OBasis‘𝑊) → 𝐵𝑉)
 
Theoremobsne0 20543 A basis element is nonzero. (Contributed by Mario Carneiro, 23-Oct-2015.)
0 = (0g𝑊)       ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴𝐵) → 𝐴0 )
 
Theoremobsocv 20544 An orthonormal basis has trivial orthocomplement. (Contributed by Mario Carneiro, 23-Oct-2015.)
0 = (0g𝑊)    &    = (ocv‘𝑊)       (𝐵 ∈ (OBasis‘𝑊) → ( 𝐵) = { 0 })
 
Theoremobs2ocv 20545 The double orthocomplement (closure) of an orthonormal basis is the whole space. (Contributed by Mario Carneiro, 23-Oct-2015.)
= (ocv‘𝑊)    &   𝑉 = (Base‘𝑊)       (𝐵 ∈ (OBasis‘𝑊) → ( ‘( 𝐵)) = 𝑉)
 
Theoremobselocv 20546 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.)
= (ocv‘𝑊)       ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶𝐵𝐴𝐵) → (𝐴 ∈ ( 𝐶) ↔ ¬ 𝐴𝐶))
 
Theoremobs2ss 20547 A basis has no proper subsets that are also bases. (Contributed by Mario Carneiro, 23-Oct-2015.)
((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ∈ (OBasis‘𝑊) ∧ 𝐶𝐵) → 𝐶 = 𝐵)
 
Theoremobslbs 20548 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.)
𝐽 = (LBasis‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)       (𝐵 ∈ (OBasis‘𝑊) → (𝐵𝐽 ↔ (𝑁𝐵) ∈ 𝐶))
 
PART 11  BASIC LINEAR ALGEBRA

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." Dealing with modules (over rings) instead of vector spaces (over fields) allows for a more unified approach. Therefore, linear equations, matrices, determinants, are usually regarded as "over a ring" in this part.

Unless otherwise stated, the rings of scalars need not be commutative (see df-cring 19421), but the existence of a multiplicative neutral element is always assumed (our rings are unital, see df-ring 19420).

For readers knowing vector spaces but unfamiliar with modules: the elements of a module are still called "vectors" and they still form a group under addition, with a zero vector as neutral element, like in a vector space. Like in a vector space, vectors can be multiplied by scalars, with the usual rules, the only difference being that the scalars are only required to form a ring, and not necessarily a field or a division ring. Note that any vector space is a (special kind of) module, so any theorem proved below for modules applies to any vector space.

 
11.1  Vectors and free modules
 
11.1.1  Direct sum of left modules

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.".
In [Lang] p. 128, the definition of the direct sum of left modules is based on direct sums of abelian groups ("We define on [the direct sum of abelian groups Mi] M a structure of A-module: If (xi)i ∈ I is an element of M, i.e. a familiy of elements xi ∈ Mi such that xi = 0 for almost all i, and if a ∈ A, then we define a(xi)i ∈ I = (axi)i ∈ I, that is we define multiplication by a componentwise.") which itself is based on the direct product of abelian groups ([Lang] p. 36: "Let {Ai}i ∈ I be a family of abelian groups. We define their direct sum A ... to be the subset of the direct product ... consisting of all families (xi)i ∈ I with xi ∈ Ai such that xi = 0 for all but a finite number of indices i").
In short, the direct sum of a familiy of (left) modules {Mi}i ∈ I is the restriction of the direct product of {Mi}i ∈ I to the elements with index function having finite support, as formalized by Definition df-dsmm 20550.

 
Syntaxcdsmm 20549 Class of module direct sum generator.
class m
 
Definitiondf-dsmm 20550* 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 ∈ V, 𝑟 ∈ V ↦ ((𝑠Xs𝑟) ↾s {𝑓X𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) ∣ {𝑥 ∈ dom 𝑟 ∣ (𝑓𝑥) ≠ (0g‘(𝑟𝑥))} ∈ Fin}))
 
Theoremreldmdsmm 20551 The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.)
Rel dom ⊕m
 
Theoremdsmmval 20552* Value of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.)
𝐵 = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))} ∈ Fin}       (𝑅𝑉 → (𝑆m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵))
 
Theoremdsmmbase 20553* Base set of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.)
𝐵 = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))} ∈ Fin}       (𝑅𝑉𝐵 = (Base‘(𝑆m 𝑅)))
 
Theoremdsmmval2 20554 Self-referential definition of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
𝐵 = (Base‘(𝑆m 𝑅))       (𝑆m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵)
 
Theoremdsmmbas2 20555* Base set of the direct sum module using the fndmin 6824 abbreviation. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝑃 = (𝑆Xs𝑅)    &   𝐵 = {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g𝑅)) ∈ Fin}       ((𝑅 Fn 𝐼𝐼𝑉) → 𝐵 = (Base‘(𝑆m 𝑅)))
 
Theoremdsmmfi 20556 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.)
((𝑅 Fn 𝐼𝐼 ∈ Fin) → (𝑆m 𝑅) = (𝑆Xs𝑅))
 
Theoremdsmmelbas 20557* Membership in the finitely supported hull of a structure product in terms of the index set. (Contributed by Stefan O'Rear, 11-Jan-2015.)
𝑃 = (𝑆Xs𝑅)    &   𝐶 = (𝑆m 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐻 = (Base‘𝐶)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 Fn 𝐼)       (𝜑 → (𝑋𝐻 ↔ (𝑋𝐵 ∧ {𝑎𝐼 ∣ (𝑋𝑎) ≠ (0g‘(𝑅𝑎))} ∈ Fin)))
 
Theoremdsmm0cl 20558 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.)
𝑃 = (𝑆Xs𝑅)    &   𝐻 = (Base‘(𝑆m 𝑅))    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Mnd)    &    0 = (0g𝑃)       (𝜑0𝐻)
 
Theoremdsmmacl 20559 The finite hull is closed under addition. (Contributed by Stefan O'Rear, 11-Jan-2015.)
𝑃 = (𝑆Xs𝑅)    &   𝐻 = (Base‘(𝑆m 𝑅))    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Mnd)    &   (𝜑𝐽𝐻)    &   (𝜑𝐾𝐻)    &    + = (+g𝑃)       (𝜑 → (𝐽 + 𝐾) ∈ 𝐻)
 
Theoremprdsinvgd2 20560 Negation of a single coordinate in a structure product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Grp)    &   𝐵 = (Base‘𝑌)    &   𝑁 = (invg𝑌)    &   (𝜑𝑋𝐵)    &   (𝜑𝐽𝐼)       (𝜑 → ((𝑁𝑋)‘𝐽) = ((invg‘(𝑅𝐽))‘(𝑋𝐽)))
 
Theoremdsmmsubg 20561 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.)
𝑃 = (𝑆Xs𝑅)    &   𝐻 = (Base‘(𝑆m 𝑅))    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Grp)       (𝜑𝐻 ∈ (SubGrp‘𝑃))
 
Theoremdsmmlss 20562* 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.)
(𝜑𝐼𝑊)    &   (𝜑𝑆 ∈ Ring)    &   (𝜑𝑅:𝐼⟶LMod)    &   ((𝜑𝑥𝐼) → (Scalar‘(𝑅𝑥)) = 𝑆)    &   𝑃 = (𝑆Xs𝑅)    &   𝑈 = (LSubSp‘𝑃)    &   𝐻 = (Base‘(𝑆m 𝑅))       (𝜑𝐻𝑈)
 
Theoremdsmmlmod 20563* 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.)
(𝜑𝐼𝑊)    &   (𝜑𝑆 ∈ Ring)    &   (𝜑𝑅:𝐼⟶LMod)    &   ((𝜑𝑥𝐼) → (Scalar‘(𝑅𝑥)) = 𝑆)    &   𝐶 = (𝑆m 𝑅)       (𝜑𝐶 ∈ LMod)
 
11.1.2  Free modules

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 nonfree 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, a free module is defined as the direct sum of copies of a ring regarded as a left module over itself, see df-frlm 20565. Since a module has a basis if and only if it is isomorphic to a free module as defined by df-frlm 20565 (see lmisfree 20660), the two definitions are essentially equivalent. The free modules as defined by df-frlm 20565 are also taken as a motivation to introduce free modules by [Lang] p. 135.

 
Syntaxcfrlm 20564 Class of free module generator.
class freeLMod
 
Definitiondf-frlm 20565* Define the function associating with a ring and a set the direct sum indexed by that set of copies of that ring regarded as a left module over itself. Recall from df-dsmm 20550 that an element of a direct sum has finitely many nonzero coordinates. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟m (𝑖 × {(ringLMod‘𝑟)})))
 
Theoremfrlmval 20566 Value of the "free module" function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)       ((𝑅𝑉𝐼𝑊) → 𝐹 = (𝑅m (𝐼 × {(ringLMod‘𝑅)})))
 
Theoremfrlmlmod 20567 The free module is a module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)       ((𝑅 ∈ Ring ∧ 𝐼𝑊) → 𝐹 ∈ LMod)
 
Theoremfrlmpws 20568 The free module as a restriction of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝐹)       ((𝑅𝑉𝐼𝑊) → 𝐹 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))
 
Theoremfrlmlss 20569 The base set of the free module is a subspace of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝐹)    &   𝑈 = (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼))       ((𝑅 ∈ Ring ∧ 𝐼𝑊) → 𝐵𝑈)
 
Theoremfrlmpwsfi 20570 The finite free module is a power of the ring module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)       ((𝑅𝑉𝐼 ∈ Fin) → 𝐹 = ((ringLMod‘𝑅) ↑s 𝐼))
 
Theoremfrlmsca 20571 The ring of scalars of a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)       ((𝑅𝑉𝐼𝑊) → 𝑅 = (Scalar‘𝐹))
 
Theoremfrlm0 20572 Zero in a free module (ring constraint is stronger than necessary, but allows use of frlmlss 20569). (Contributed by Stefan O'Rear, 4-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑊) → (𝐼 × { 0 }) = (0g𝐹))
 
Theoremfrlmbas 20573* Base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 23-Jun-2019.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝑁 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐵 = {𝑘 ∈ (𝑁m 𝐼) ∣ 𝑘 finSupp 0 }       ((𝑅𝑉𝐼𝑊) → 𝐵 = (Base‘𝐹))
 
Theoremfrlmelbas 20574 Membership in the base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 23-Jun-2019.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝑁 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐵 = (Base‘𝐹)       ((𝑅𝑉𝐼𝑊) → (𝑋𝐵 ↔ (𝑋 ∈ (𝑁m 𝐼) ∧ 𝑋 finSupp 0 )))
 
Theoremfrlmrcl 20575 If a free module is inhabited, this is sufficient to conclude that the ring expression defines a set. (Contributed by Stefan O'Rear, 3-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝐹)       (𝑋𝐵𝑅 ∈ V)
 
Theoremfrlmbasfsupp 20576 Elements of the free module are finitely supported. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Revised by Thierry Arnoux, 21-Jun-2019.) (Proof shortened by AV, 20-Jul-2019.)
𝐹 = (𝑅 freeLMod 𝐼)    &    0 = (0g𝑅)    &   𝐵 = (Base‘𝐹)       ((𝐼𝑊𝑋𝐵) → 𝑋 finSupp 0 )
 
Theoremfrlmbasmap 20577 Elements of the free module are set functions. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝑁 = (Base‘𝑅)    &   𝐵 = (Base‘𝐹)       ((𝐼𝑊𝑋𝐵) → 𝑋 ∈ (𝑁m 𝐼))
 
Theoremfrlmbasf 20578 Elements of the free module are functions. (Contributed by Stefan O'Rear, 3-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝑁 = (Base‘𝑅)    &   𝐵 = (Base‘𝐹)       ((𝐼𝑊𝑋𝐵) → 𝑋:𝐼𝑁)
 
Theoremfrlmlvec 20579 The free module over a division ring is a left vector space. (Contributed by Steven Nguyen, 29-Apr-2023.)
𝐹 = (𝑅 freeLMod 𝐼)       ((𝑅 ∈ DivRing ∧ 𝐼𝑊) → 𝐹 ∈ LVec)
 
Theoremfrlmfibas 20580 The base set of the finite free module as a set exponential. (Contributed by AV, 6-Dec-2018.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝑁 = (Base‘𝑅)       ((𝑅𝑉𝐼 ∈ Fin) → (𝑁m 𝐼) = (Base‘𝐹))
 
Theoremelfrlmbasn0 20581 If the dimension of a free module over a ring is not 0, every element of its base set is not empty. (Contributed by AV, 10-Feb-2019.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝑁 = (Base‘𝑅)    &   𝐵 = (Base‘𝐹)       ((𝐼𝑉𝐼 ≠ ∅) → (𝑋𝐵𝑋 ≠ ∅))
 
Theoremfrlmplusgval 20582 Addition in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑅𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &    + = (+g𝑅)    &    = (+g𝑌)       (𝜑 → (𝐹 𝐺) = (𝐹f + 𝐺))
 
Theoremfrlmsubgval 20583 Subtraction in a free module. (Contributed by Thierry Arnoux, 30-Jun-2019.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝑊)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &    = (-g𝑅)    &   𝑀 = (-g𝑌)       (𝜑 → (𝐹𝑀𝐺) = (𝐹f 𝐺))
 
Theoremfrlmvscafval 20584 Scalar multiplication in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑌)    &   𝐾 = (Base‘𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝐴𝐾)    &   (𝜑𝑋𝐵)    &    = ( ·𝑠𝑌)    &    · = (.r𝑅)       (𝜑 → (𝐴 𝑋) = ((𝐼 × {𝐴}) ∘f · 𝑋))
 
Theoremfrlmvplusgvalc 20585 Coordinates of a sum with respect to a basis in a free module. (Contributed by AV, 16-Jan-2023.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝐹)    &   (𝜑𝑅𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐽𝐼)    &    + = (+g𝑅)    &    = (+g𝐹)       (𝜑 → ((𝑋 𝑌)‘𝐽) = ((𝑋𝐽) + (𝑌𝐽)))
 
Theoremfrlmvscaval 20586 Coordinates of a scalar multiple with respect to a basis in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑌)    &   𝐾 = (Base‘𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝐴𝐾)    &   (𝜑𝑋𝐵)    &   (𝜑𝐽𝐼)    &    = ( ·𝑠𝑌)    &    · = (.r𝑅)       (𝜑 → ((𝐴 𝑋)‘𝐽) = (𝐴 · (𝑋𝐽)))
 
Theoremfrlmplusgvalb 20587* Addition in a free module at the coordinates. (Contributed by AV, 16-Jan-2023.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝐹)    &   (𝜑𝐼𝑊)    &   (𝜑𝑋𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑌𝐵)    &    + = (+g𝑅)    &    = (+g𝐹)       (𝜑 → (𝑍 = (𝑋 𝑌) ↔ ∀𝑖𝐼 (𝑍𝑖) = ((𝑋𝑖) + (𝑌𝑖))))
 
Theoremfrlmvscavalb 20588* Scalar multiplication in a free module at the coordinates. (Contributed by AV, 16-Jan-2023.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝐹)    &   (𝜑𝐼𝑊)    &   (𝜑𝑋𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑅 ∈ Ring)    &   𝐾 = (Base‘𝑅)    &   (𝜑𝐴𝐾)    &    = ( ·𝑠𝐹)    &    · = (.r𝑅)       (𝜑 → (𝑍 = (𝐴 𝑋) ↔ ∀𝑖𝐼 (𝑍𝑖) = (𝐴 · (𝑋𝑖))))
 
Theoremfrlmvplusgscavalb 20589* Addition combined with scalar multiplication in a free module at the coordinates. (Contributed by AV, 16-Jan-2023.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝐹)    &   (𝜑𝐼𝑊)    &   (𝜑𝑋𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑅 ∈ Ring)    &   𝐾 = (Base‘𝑅)    &   (𝜑𝐴𝐾)    &    = ( ·𝑠𝐹)    &    · = (.r𝑅)    &   (𝜑𝑌𝐵)    &    + = (+g𝑅)    &    = (+g𝐹)    &   (𝜑𝐶𝐾)       (𝜑 → (𝑍 = ((𝐴 𝑋) (𝐶 𝑌)) ↔ ∀𝑖𝐼 (𝑍𝑖) = ((𝐴 · (𝑋𝑖)) + (𝐶 · (𝑌𝑖)))))
 
Theoremfrlmgsum 20590* Finite commutative sums in a free module are taken componentwise. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Mario Carneiro, 5-Jul-2015.) (Revised by AV, 23-Jun-2019.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑌)    &    0 = (0g𝑌)    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝑊)    &   (𝜑𝑅 ∈ Ring)    &   ((𝜑𝑦𝐽) → (𝑥𝐼𝑈) ∈ 𝐵)    &   (𝜑 → (𝑦𝐽 ↦ (𝑥𝐼𝑈)) finSupp 0 )       (𝜑 → (𝑌 Σg (𝑦𝐽 ↦ (𝑥𝐼𝑈))) = (𝑥𝐼 ↦ (𝑅 Σg (𝑦𝐽𝑈))))
 
Theoremfrlmsplit2 20591* Restriction is homomorphic on free modules. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
𝑌 = (𝑅 freeLMod 𝑈)    &   𝑍 = (𝑅 freeLMod 𝑉)    &   𝐵 = (Base‘𝑌)    &   𝐶 = (Base‘𝑍)    &   𝐹 = (𝑥𝐵 ↦ (𝑥𝑉))       ((𝑅 ∈ Ring ∧ 𝑈𝑋𝑉𝑈) → 𝐹 ∈ (𝑌 LMHom 𝑍))
 
Theoremfrlmsslss 20592* A subset of a free module obtained by restricting the support set is a submodule. 𝐽 is the set of forbidden unit vectors. (Contributed by Stefan O'Rear, 4-Feb-2015.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝑈 = (LSubSp‘𝑌)    &   𝐵 = (Base‘𝑌)    &    0 = (0g𝑅)    &   𝐶 = {𝑥𝐵 ∣ (𝑥𝐽) = (𝐽 × { 0 })}       ((𝑅 ∈ Ring ∧ 𝐼𝑉𝐽𝐼) → 𝐶𝑈)
 
Theoremfrlmsslss2 20593* A subset of a free module obtained by restricting the support set is a submodule. 𝐽 is the set of permitted unit vectors. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 23-Jun-2019.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝑈 = (LSubSp‘𝑌)    &   𝐵 = (Base‘𝑌)    &    0 = (0g𝑅)    &   𝐶 = {𝑥𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽}       ((𝑅 ∈ Ring ∧ 𝐼𝑉𝐽𝐼) → 𝐶𝑈)
 
Theoremfrlmbas3 20594 An element of the base set of a finite free module with a Cartesian product as index set as operation value. (Contributed by AV, 14-Feb-2019.)
𝐹 = (𝑅 freeLMod (𝑁 × 𝑀))    &   𝐵 = (Base‘𝑅)    &   𝑉 = (Base‘𝐹)       (((𝑅𝑊𝑋𝑉) ∧ (𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ (𝐼𝑁𝐽𝑀)) → (𝐼𝑋𝐽) ∈ 𝐵)
 
Theoremmpofrlmd 20595* Elements of the free module are mappings with two arguments defined by their operation values. (Contributed by AV, 20-Feb-2019.) (Proof shortened by AV, 3-Jul-2022.)
𝐹 = (𝑅 freeLMod (𝑁 × 𝑀))    &   𝑉 = (Base‘𝐹)    &   ((𝑖 = 𝑎𝑗 = 𝑏) → 𝐴 = 𝐵)    &   ((𝜑𝑖𝑁𝑗𝑀) → 𝐴𝑋)    &   ((𝜑𝑎𝑁𝑏𝑀) → 𝐵𝑌)    &   (𝜑 → (𝑁𝑈𝑀𝑊𝑍𝑉))       (𝜑 → (𝑍 = (𝑎𝑁, 𝑏𝑀𝐵) ↔ ∀𝑖𝑁𝑗𝑀 (𝑖𝑍𝑗) = 𝐴))
 
Theoremfrlmip 20596* The inner product of a free module. (Contributed by Thierry Arnoux, 20-Jun-2019.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝐼𝑊𝑅𝑉) → (𝑓 ∈ (𝐵m 𝐼), 𝑔 ∈ (𝐵m 𝐼) ↦ (𝑅 Σg (𝑥𝐼 ↦ ((𝑓𝑥) · (𝑔𝑥))))) = (·𝑖𝑌))
 
Theoremfrlmipval 20597 The inner product of a free module. (Contributed by Thierry Arnoux, 21-Jun-2019.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑉 = (Base‘𝑌)    &    , = (·𝑖𝑌)       (((𝐼𝑊𝑅𝑋) ∧ (𝐹𝑉𝐺𝑉)) → (𝐹 , 𝐺) = (𝑅 Σg (𝐹f · 𝐺)))
 
Theoremfrlmphllem 20598* Lemma for frlmphl 20599. (Contributed by AV, 21-Jul-2019.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑉 = (Base‘𝑌)    &    , = (·𝑖𝑌)    &   𝑂 = (0g𝑌)    &    0 = (0g𝑅)    &    = (*𝑟𝑅)    &   (𝜑𝑅 ∈ Field)    &   ((𝜑𝑔𝑉 ∧ (𝑔 , 𝑔) = 0 ) → 𝑔 = 𝑂)    &   ((𝜑𝑥𝐵) → ( 𝑥) = 𝑥)    &   (𝜑𝐼𝑊)       ((𝜑𝑔𝑉𝑉) → (𝑥𝐼 ↦ ((𝑔𝑥) · (𝑥))) finSupp 0 )
 
Theoremfrlmphl 20599* Conditions for a free module to be a pre-Hilbert space. (Contributed by Thierry Arnoux, 21-Jun-2019.) (Proof shortened by AV, 21-Jul-2019.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑉 = (Base‘𝑌)    &    , = (·𝑖𝑌)    &   𝑂 = (0g𝑌)    &    0 = (0g𝑅)    &    = (*𝑟𝑅)    &   (𝜑𝑅 ∈ Field)    &   ((𝜑𝑔𝑉 ∧ (𝑔 , 𝑔) = 0 ) → 𝑔 = 𝑂)    &   ((𝜑𝑥𝐵) → ( 𝑥) = 𝑥)    &   (𝜑𝐼𝑊)       (𝜑𝑌 ∈ PreHil)
 
11.1.3  Standard basis (unit vectors)

According to Wikipedia ("Standard basis", 16-Mar-2019, https://en.wikipedia.org/wiki/Standard_basis) "In mathematics, the standard basis (also called natural basis) for a Euclidean space is the set of unit vectors pointing in the direction of the axes of a Cartesian coordinate system.", and ("Unit vector", 16-Mar-2019, https://en.wikipedia.org/wiki/Unit_vector) "In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1.". In the following, the term "unit vector" (or more specific "basic unit vector") is used for the (special) unit vectors forming the standard basis of free modules. However, the length of the unit vectors is not considered here, so it is not required to regard normed spaces.

 
Syntaxcuvc 20600 Class of basic unit vectors for an explicit free module.
class unitVec
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