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{{Traducció|en|Quotient space (linear algebra)}}
In LINEAR ALGEBRA , the ((((( quotient ))))) of a VECTOR SPACE (( V )) by a SUBSPACE (( N )) is a vector space obtained by "collapsing" (( N )) to zero. The space obtained is called a ((((( quotient space ))))) and is denoted (( V )) /(( N )) (read (( V )) mod (( N )) ). ..
In [[linear algebra]], the '''quotient''' of a [[vector space]] ''V'' by a [[linear subspace|subspace]] ''N'' is a vector space obtained by "collapsing" ''N'' to zero. The space obtained is called a '''quotient space''' and is denoted ''V''/''N'' (read ''V'' mod ''N'').
..
 
(==) Definition (==) ..
En L'[[àlgebra lineal]], el '''quocient''' d'un [[espai vectorial]] ''V'' al costat d'un [[subespai]] ''N'' és "col·lapsant" un espai vectorial obtingut a prop ''N'' a zero. L'espai obtingut es crida un '''espai quocient''' i és denotat ''V'' /''N'' (lectura ''V'' mod ''N'' ).
Formally , the construction is as follows {{ harv .3. Halmos .3. 1974 .3. loc=§21-22}}. Let (( V )) be a VECTOR SPACE over a FIELD (( K )) , and let (( N )) be a SUBSPACE of (( V )) . We define an EQUIVALENCE RELATION ~ on (( V )) by stating that (( x )) ~ (( y )) if (( x )) &nbsp ;&minus ;&nbsp ;(( y )) &isin ; (( N )) . That is , (( x )) is related to (( y )) if one can be obtained from the other by adding an element of (( N )) . From this definition , one can deduce that any element of (( N )) is equivalent to the zero vector ; in other words all the vectors in (( N )) get mapped into the equivalence class of the zero vector. ..
 
..
 
The EQUIVALENCE CLASS of (( x )) is often denoted ..
 
: [(( x )) ] = (( x )) + (( N )) ..
== Definition ==
since it is given by ..
 
: [(( x )) ] = {(( x )) + (( n )) : (( n )) &isin ; (( N )) }. ..
== Definició ==
..
Formally, the construction is as follows {{harv|Halmos|1974|loc=§21-22}}. Let ''V'' be a [[vector space]] over a [[field (mathematics)|field]] ''K'', and let ''N'' be a [[linear subspace|subspace]] of ''V''. We define an [[equivalence relation]] ~ on ''V'' by stating that ''x'' ~ ''y'' if ''x'' − ''y'' ∈ ''N''. That is, ''x'' is related to ''y'' if one can be obtained from the other by adding an element of ''N''. From this definition, one can deduce that any element of ''N'' is equivalent to the zero vector; in other words all the vectors in ''N'' get mapped into the equivalence class of the zero vector.
The quotient space (( V )) /(( N )) is then defined as (( V )) /~ , the set of all equivalence classes over (( V )) by ~. Scalar multiplication and addition are defined on the equivalence classes by ..
 
* &alpha ;[(( x )) ] = [&alpha ;(( x )) ] for all &alpha ; &isin ; (( K )) , and ..
Formalment, la construcció és de la manera següent {{harv|Halmos|1974|loc=§21-22}}. Sia ''V'' un [[espai vectorial]] sobre un [[cos (matemàtiques)|camp]] ''K'', i deixar ''N'' ser un [[subespai]] de ''V'' . Definim una [[relació d'equivalència]] ~ damunt ''V'' manifestant allò ''x'' ~ ''y'' si ''x''  − ''y'' ∈ ''N'' . És a dir ''x'' és referit a ''y'' si un se'n pot obtenir des de l'altre afegint un element de ''N'' . D'aquesta definició, un pot deduir allò qualsevol element de ''N'' és equivalent al zero vector; en altres paraules tots els vectors en ''N'' ser correspost a la classe d'equivalència del zero vector.
* [(( x )) ]&nbsp ;+&nbsp ;[(( y )) ] = [(( x )) +(( y )) ]. ..
 
It is not hard to check that these operations are WELL-DEFINED (i.e. do not depend on the choice of representative). These operations turn the quotient space (( V )) /(( N )) into a vector space over (( K )) with (( N )) being the zero class , [0]. ..
 
..
 
The mapping that associates to (( v )) &nbsp ;&isin ;&nbsp ;(( V )) the equivalence class [(( v )) ] is known as the ((((( quotient map ))))). ..
The [[equivalence class]] of ''x'' is often denoted
..
 
(==) Examples (==) ..
La [[classe d'equivalència]] de ''x'' és sovint denotat
Let (( X )) &nbsp ;=&nbsp ;((((( R )))))<sup>2</sup> be the standard Cartesian plane , and let (( Y )) be a line through the origin in (( X )) . Then the quotient space (( X )) /(( Y )) can be identified with the space of all lines in (( X )) which are parallel to (( Y )) . That is to say that , the elements of the set (( X )) /(( Y )) are lines in (( X )) parallel to (( Y )) . This gives one way in which to visualize quotient spaces geometrically. ..
:[''x''] = ''x'' + ''N''
..
 
Another example is the quotient of ((((( R )))))<sup>(( n )) </sup> by the subspace spanned by the first (( m )) standard basis vectors. The space ((((( R )))))<sup>(( n )) </sup> consists of all (( n )) -tuples of real numbers ((( x )) <sub>1</sub> ,&#8230; ,(( x )) <sub>(( n )) </sub>). The subspace , identified with ((((( R )))))<sup>(( m )) </sup> , consists of all (( n )) -tuples such that only the first (( m )) entries are non-zero: ((( x )) <sub>1</sub> ,&#8230; ,(( x )) <sub>(( m )) </sub> ,0 ,0 ,&#8230; ,0). Two vectors of ((((( R )))))<sup>(( n )) </sup> are in the same congruence class modulo the subspace if and only if they are identical in the last (( n )) &minus ;(( m )) coordinates. The quotient space ((((( R )))))<sup>(( n )) </sup>/ ((((( R )))))<sup>(( m )) </sup> is ISOMORPHIC to ((((( R )))))<sup>(( n )) &minus ;(( m )) </sup> in an obvious manner. ..
: [''x'' ] = ''x'' + ''N''
..
since it is given by
More generally , if (( V )) is an (internal) DIRECT SUM of subspaces (( U )) and (( W )) : ..
 
: <math>V=U\oplus W</math> ..
ja que es dóna a prop
then the quotient space (( V )) /(( U )) is naturally isomorphic to (( W )) {{ harv .3. Halmos .3. 1974 .3. loc=Theorem 22.1}}. ..
:[''x''] = {''x'' + ''n'' : ''n'' &isin; ''N''}.
..
 
(==) Properties (==) ..
: [''x'' ] = {''x'' + ''n'' : ''n'' ∈ ''N'' }.
..
 
There is a natural EPIMORPHISM from (( V )) to the quotient space (( V )) /(( U )) given by sending (( x )) to its equivalence class [(( x )) ]. The KERNEL (or NULLSPACE) of this epimorphism is the subspace (( U )) . This relationship is neatly summarized by the SHORT EXACT SEQUENCE ..
 
: <math>0\to U\to V\to V/U\to 0.\ ,</math> ..
 
..
The quotient space ''V''/''N'' is then defined as ''V''/~, the set of all equivalence classes over ''V'' by ~. Scalar multiplication and addition are defined on the equivalence classes by
If (( U )) is a subspace of (( V )) , the DIMENSION of (( V )) /(( U )) is called the ((((( CODIMENSION ))))) of (( U )) in (( V )) . Since a basis of (( V )) may be constructed from a basis (( A )) of (( U )) and a basis (( B )) of (( V )) /(( U )) by adding a representative of each element of (( B )) to (( A )) , the dimension of (( V )) is the sum of the dimensions of (( U )) and (( V )) /(( U )) . If (( V )) is FINITE-DIMENSIONAL , it follows that the codimension of (( U )) in (( V )) is the difference between the dimensions of (( V )) and (( U )) {{ harv .3. Halmos .3. 1974 .3. loc=Theorem 22.2}}: ..
 
: <math>\mathrm{codim}(U) = \dim(V/U) = \dim(V) - \dim(U).</math> ..
L'espai quocient ''V'' /''N'' és llavors definit com ''V'' /~, el conjunt de tota l'equivalència classifica per damunt ''V'' per ~. La multiplicació escalar i addició es defineixen en les classes d'equivalència a prop
..
*&alpha;[''x''] = [&alpha;''x''] for all &alpha; &isin; ''K'', and
Let (( T )) : (( V )) &rarr ; (( W )) be a LINEAR OPERATOR. The kernel of (( T )) , denoted ker(((( T )))) , is the set of all (( x )) &isin ; (( V )) such that (( Tx )) = 0. The kernel is a subspace of (( V )) . The FIRST ISOMORPHISM THEOREM of linear algebra says that the quotient space (( V )) /ker(((( T )))) is isomorphic to the image of (( V )) in (( W )) . An immediate corollary , for finite-dimensional spaces , is the RANK-NULLITY THEOREM: the dimension of (( V )) is equal to the dimension of the kernel (the (( nullity )) of (( T )) ) plus the dimension of the image (the (( rank )) of (( T )) ). ..
 
..
* α;[''x'' ] = [α;''x'' ] per a tot el α; ∈ ''K'', i
The COKERNEL of a linear operator (( T )) : (( V )) &rarr ; (( W )) is defined to be the quotient space (( W )) /im(((( T )))) . ..
*[''x'']&nbsp;+&nbsp;[''y''] = [''x''+''y''].
..
 
(==) Quotient of a Banach space by a subspace (==) ..
* [''x'' ] + [''y'' ] = [''x'' +''y'' ].
If (( X )) is a BANACH SPACE and (( M )) is a CLOSED subspace of (( X )) , then the quotient (( X )) /(( M )) is again a Banach space. The quotient space is already endowed with a vector space structure by the construction of the previous section. We define a norm on (( X )) /(( M )) by ..
It is not hard to check that these operations are [[well-defined]] (i.e. do not depend on the choice of representative). These operations turn the quotient space ''V''/''N'' into a vector space over ''K'' with ''N'' being the zero class, [0].
: <math> \ .3. [x] \ .3. _{X/M} = \inf_{m \in M} \ .3. x-m\ .3. _X. </math> ..
 
The quotient space (( X )) /(( M )) is COMPLETE with respect to the norm , so it is a Banach space. ..
No és dur d'aturar que aquestes operacions siguin [[well-defined]] (i.e. no depengui de l'elecció de representant). Aquestes operacions giren l'espai quocient ''V'' /''N'' a un espai vectorial sobre ''K'' amb ''N'' sent la zero classe, [0].
..
 
(===) Examples (===) ..
 
Let (( C )) [0 ,1] denote the Banach space of continuous real-valued functions on the interval [0 ,1] with the SUP NORM. Denote the subspace of all functions (( f )) &isin ; (( C )) [0 ,1] with (( f )) (0) = 0 by (( M )) . Then the equivalence class of some function (( g )) is determined by its value at 0 , and the quotient space (( C )) [0 ,1]&nbsp ;/&nbsp ;(( M )) is isomorphic to ((((( R ))))). ..
 
..
The mapping that associates to ''v''&nbsp;&isin;&nbsp;''V'' the equivalence class [''v''] is known as the '''quotient map'''.
If (( X )) is a HILBERT SPACE , then the quotient space (( X )) /(( M )) is isomorphic to the HILBERT SPACE#ORTHOGONAL COMPLEMENTS AND PROJECTIONS .3. ORTHOGONAL COMPLEMENT .3. orthogonal complement]] of (( M )) . ..
 
..
El mapatge que s'associa a ''v''  ∈ ''V'' la classe d'equivalència [''v'' ] és sabut com el '''mapa de quocient'''.
(===) Generalization to locally convex spaces (===) ..
 
The quotient of a LOCALLY CONVEX SPACE by a closed subspace is again locally convex {{ harv .3. Dieudonné .3. 1970 .3. loc=12.14.8}}. Indeed , suppose that (( X )) is locally convex so that the topology on (( X )) is generated by a family of SEMINORMS {(( p )) <sub>&alpha ;</sub> .3. &alpha ;&isin ;(( A )) } where (( A )) is an index set. Let (( M )) be a closed subspace , and define seminorms (( q )) <sub>&alpha</sub> by on (( X )) /(( M )) ..
 
..
 
: <math>q_\alpha([x]) = \inf_{x\in [x]} p_\alpha(x).</math> ..
== Examples ==
..
 
Then (( X )) /(( M )) is a locally convex space , and the topology on it is the QUOTIENT TOPOLOGY. ..
== Exemples ==
..
Let ''X''&nbsp;=&nbsp;'''R'''<sup>2</sup> be the standard Cartesian plane, and let ''Y'' be a line through the origin in ''X''. Then the quotient space ''X''/''Y'' can be identified with the space of all lines in ''X'' which are parallel to ''Y''. That is to say that, the elements of the set ''X''/''Y'' are lines in ''X'' parallel to ''Y''. This gives one way in which to visualize quotient spaces geometrically.
If , furthermore , (( X )) is METRIZABLE , then so is (( X )) /(( M )) . If (( X )) is a FRÉCHET SPACE , then so is (( X )) /(( M )) {{ harv .3. Dieudonné .3. 1970 .3. loc=12.11.3}}. ..
 
..
Sia ''X''  = '''R'''<sup>2</sup> el pla cartesià estàndard, i deixar ''Y'' ser una línia a través de l'origen dins ''X'' . Llavors l'espai quocient ''X'' /''Y'' pot ser identificat amb l'espai de totes les línies en ''X'' que són paral·lels a ''Y'' . És a dir que, els elements del conjunt ''X'' /''Y'' són línies en ''X'' paral·lel a ''Y'' . Això entrega un camí quin visualitzar espais quocient geomètricament.
(==) See also (==) ..
 
* QUOTIENT SET ..
 
* QUOTIENT GROUP ..
 
* QUOTIENT MODULE ..
Another example is the quotient of '''R'''<sup>''n''</sup> by the subspace spanned by the first ''m'' standard basis vectors. The space '''R'''<sup>''n''</sup> consists of all ''n''-tuples of real numbers (''x''<sub>1</sub>,…,''x''<sub>''n''</sub>). The subspace, identified with '''R'''<sup>''m''</sup>, consists of all ''n''-tuples such that only the first ''m'' entries are non-zero: (''x''<sub>1</sub>,…,''x''<sub>''m''</sub>,0,0,…,0). Two vectors of '''R'''<sup>''n''</sup> are in the same congruence class modulo the subspace if and only if they are identical in the last ''n''&minus;''m'' coordinates. The quotient space '''R'''<sup>''n''</sup>/ '''R'''<sup>''m''</sup> is [[isomorphic]] to '''R'''<sup>''n''&minus;''m''</sup> in an obvious manner.
* QUOTIENT SPACE (in TOPOLOGY) ..
 
..
Un altre exemple és el quocient de '''R'''<sup>''n'' </sup> al costat del subespai abraçat abans del primer ''m'' vectors de base estàndards. L'espai '''R'''<sup>''n'' </sup> consisteix de tot ''n'' -tuples de nombres reals ''( x'' <sub>1</sub>,…;,''x'' <sub>''n'' </sub>). El subespai, identificava amb '''R'''<sup>''m'' </sup>, consta de tot ''n'' -tuples tal que només el primer ''m'' entrades són non-zero: ''( x'' <sub>1</sub>,…;,''x'' <sub>''m'' </sub>,0,0,…;,0). Dos vectors de '''R'''<sup>''n'' </sup> són en el mateix modulo de classe de congruència el subespai si i només si són idèntics en l'últim ''n'' −''m'' coordenades. L'espai quocient '''R'''<sup>''n'' </sup>/ '''R'''<sup>''m'' </sup> és [[isomorfisme|isomorf]] a '''R'''<sup>''n'' −''m'' </sup> en una conducta òbvia.
(==) References (==) ..
 
* {{ citation .3. first=Paul .3. last=Halmos .3. authorlink=Paul Halmos .3. title=Finite dimensional vector spaces .3. publisher=Springer .3. year=1974 .3. isbn=978-0387900933}}. ..
 
* {{ citation .3. first=Jean .3. last=Dieudonné .3. authorlink=Jean Dieudonné .3. title=Treatise on analysis , Volume Ii .3. publisher=Academic Press .3. year=1970}}. ..
 
..
More generally, if ''V'' is an (internal) [[direct sum of vector spaces|direct sum]] of subspaces ''U'' and ''W'':
[[Category:Linear algebra]] ..
 
[[Category:Functional analysis]] ..
Més generalment, si ''V'' és un (intern) [[suma directa]] de subespais ''U'' i ''W'' :
..
:<math>V=U\oplus W</math>
[[de:Faktorraum]] ..
then the quotient space ''V''/''U'' is naturally isomorphic to ''W'' {{harv|Halmos|1974|loc=Theorem 22.1}}.
[[it:Spazio vettoriale quoziente]] ..
 
[[he:מרחב מנה (אלגברה לינארית)]] ..
llavors l'espai quocient ''V'' /''U'' és naturalment isomorf a ''W'' {{harv|Halmos|1974|loc=Theorem 22.1}}.
[[pl:Przestrzeń ilorazowa (algebra liniowa)]] ..
 
[[ru:Факторпространство по подпространству]] ..
 
[[zh:商空间 (线性代数)]] ..
 
paraulesenllacos ..
== Properties ==
..
 
LINEAR ALGEBRA ..
== Propietats ==
..
 
VECTOR SPACE ..
 
..
 
SUBSPACE ..
There is a natural [[epimorphism]] from ''V'' to the quotient space ''V''/''U'' given by sending ''x'' to its equivalence class [''x'']. The [[kernel (algebra)|kernel]] (or [[nullspace]]) of this epimorphism is the subspace ''U''. This relationship is neatly summarized by the [[short exact sequence]]
..
 
VECTOR SPACE ..
Hi ha un [[epimorfisme]] natural de ''V'' a l'espai quocient ''V'' /''U'' donat enviant ''x'' a la seva classe d'equivalència [''x'' ]. El [[nucli (matemàtiques)|nucli]] (o [[nullspace)]] d'aquest epimorfisme és el subespai ''U'' . Aquesta relació és polidament resumida per la [[seqüència exacta curta]]
..
:<math>0\to U\to V\to V/U\to 0.\,</math>
FIELD ..
 
..
 
SUBSPACE ..
 
..
If ''U'' is a subspace of ''V'', the [[dimension (vector space)|dimension]] of ''V''/''U'' is called the '''[[codimension]]''' of ''U'' in ''V''. Since a basis of ''V'' may be constructed from a basis ''A'' of ''U'' and a basis ''B'' of ''V''/''U'' by adding a representative of each element of ''B'' to ''A'', the dimension of ''V'' is the sum of the dimensions of ''U'' and ''V''/''U''. If ''V'' is [[finite-dimensional]], it follows that the codimension of ''U'' in ''V'' is the difference between the dimensions of ''V'' and ''U'' {{harv|Halmos|1974|loc=Theorem 22.2}}:
EQUIVALENCE RELATION ..
 
..
Si ''U'' és un subespai de ''V'', la [[dimensió d'un espai vectorial|dimensió]] de ''V'' /''U'' és anomenat el '''[[codimensió]]''' de ''U'' en ''V'' . Des d'una base de ''V'' pot ser construït des d'una base ''A'' de ''U'' i una base ''B'' de ''V'' /''U'' afegint un representant de cada element de ''B'' a ''A'', la dimensió de ''V'' és la suma de les dimensions de ''U'' i ''V'' /''U'' . Si ''V'' és [[dimensió d'un espai vectorial|finit dimensional]], segueix que la codimensió de ''U'' en ''V'' és la diferència entre les dimensions de ''V'' i ''U'' {{harv|Halmos|1974|loc=Theorem 22.2}}:
EQUIVALENCE CLASS ..
:<math>\mathrm{codim}(U) = \dim(V/U) = \dim(V) - \dim(U).</math>
..
 
WELL-DEFINED ..
 
..
 
ISOMORPHIC ..
Let ''T'' : ''V'' &rarr; ''W'' be a [[linear operator]]. The kernel of ''T'', denoted ker(''T''), is the set of all ''x'' &isin; ''V'' such that ''Tx'' = 0. The kernel is a subspace of ''V''. The [[first isomorphism theorem]] of linear algebra says that the quotient space ''V''/ker(''T'') is isomorphic to the image of ''V'' in ''W''. An immediate corollary, for finite-dimensional spaces, is the [[rank-nullity theorem]]: the dimension of ''V'' is equal to the dimension of the kernel (the ''nullity'' of ''T'') plus the dimension of the image (the ''rank'' of ''T'').
..
 
DIRECT SUM ..
Sia ''T'' : ''V'' → ''W'' un [[operador lineal]]. El nucli de ''T'', denotat ker(''T''), és el conjunt de tot ''x'' ∈ ''V'' tal que ''Tx'' = 0. El nucli és un subespai de ''V'' . El [[primer teorema d'isomorfisme]] d'àlgebra lineal diu que l'espai quocient ''V'' /ker(''T'') és isomorf a la imatge de ''V'' en ''W'' . Un corol·lari immediat, per a espais finits dimensionals, és el [[teorema]] de NUL·LITAT de FILA: la dimensió de ''V'' és igual a la dimensió del nucli (el ''nul·litat'' de ''T'' ) més la dimensió de la imatge (el ''fila'' de ''T'' ).
..
 
EPIMORPHISM ..
 
..
 
KERNEL ..
The [[cokernel]] of a linear operator ''T'' : ''V'' &rarr; ''W'' is defined to be the quotient space ''W''/im(''T'').
..
 
NULLSPACE ..
El [[conucli]] d'un operador lineal ''T'' : ''V'' → ''W'' és definit ser l'espai quocient ''W'' /im(''T'') .
..
 
SHORT EXACT SEQUENCE ..
 
..
 
DIMENSION ..
== Quotient of a Banach space by a subspace ==
..
 
CODIMENSION ..
== Quocient d'un espai Banach per un subespai ==
..
If ''X'' is a [[Banach space]] and ''M'' is a [[closed set|closed]] subspace of ''X'', then the quotient ''X''/''M'' is again a Banach space. The quotient space is already endowed with a vector space structure by the construction of the previous section. We define a norm on ''X''/''M'' by
FINITE-DIMENSIONAL ..
 
..
Si ''X'' és un [[Espai de Banach|Espai de]] BANACH i ''M'' és un subespai [[conjunt tancat|tancat]] de ''X'', llavors el quocient ''X'' /''M'' és una altra vegada un espai Banach. Ja dota l'espai quocient d'una estructura espacial vectorial la construcció de la secció prèvia. Definim una norma damunt ''X'' /''M'' per
LINEAR OPERATOR ..
:<math> \| [x] \|_{X/M} = \inf_{m \in M} \|x-m\|_X. </math>
..
The quotient space ''X''/''M'' is [[complete space|complete]] with respect to the norm, so it is a Banach space.
FIRST ISOMORPHISM THEOREM ..
 
..
L'espai quocient ''X'' /''M'' és [[complet]] respecte a la norma, així és un espai Banach.
RANK-NULLITY THEOREM ..
 
..
 
COKERNEL ..
 
..
=== Examples ===
BANACH SPACE ..
 
..
=== Exemples ===
CLOSED ..
Let ''C''[0,1] denote the Banach space of continuous real-valued functions on the interval [0,1] with the [[sup norm]]. Denote the subspace of all functions ''f'' &isin; ''C''[0,1] with ''f''(0) = 0 by ''M''. Then the equivalence class of some function ''g'' is determined by its value at 0, and the quotient space ''C''[0,1]&nbsp;/&nbsp;''M'' is isomorphic to '''R'''.
..
 
COMPLETE ..
Deixi ''C'' [0,1] denotar l'espai Banach de funcions genuïnament valorades contínues en l'interval [0,1] amb el [[sup norma]]. Denoti el subespai de totes les funcions ''f'' ∈ ''C'' [0,1] amb ''f'' (0) = 0 per ''M'' . Llavors la classe d'equivalència d'alguna funció ''g'' està determinat al costat del seu valor a les 0, i l'espai quocient ''C'' [0,1] / ''M'' és isomorf a '''R'''.
..
 
SUP NORM ..
 
..
 
HILBERT SPACE ..
If ''X'' is a [[Hilbert space]], then the quotient space ''X''/''M'' is isomorphic to the [[Hilbert space#Orthogonal complements and projections|orthogonal complement]] of ''M''.
..
 
HILBERT SPACE#ORTHOGONAL COMPLEMENTS AND PROJECTIONS .3. ORTHOGONAL COMPLEMENT ..
Si ''X'' és un [[Espai de Hilbert|Espai de hilbert]], llavors l'espai quocient ''X'' /''M'' és isomorf als [[Espai de Hilbert|Complements d'hilbert space#orthogonal i projeccions|complement ortogonal]]|COMPLEMENT ORTOGONAL|complement ortogonal]] de ''M'' .
..
 
LOCALLY CONVEX SPACE ..
 
..
 
SEMINORMS ..
=== Generalization to locally convex spaces ===
..
 
QUOTIENT TOPOLOGY ..
=== Generalització a espais localment convexos ===
..
The quotient of a [[locally convex space]] by a closed subspace is again locally convex {{harv|Dieudonné|1970|loc=12.14.8}}. Indeed, suppose that ''X'' is locally convex so that the topology on ''X'' is generated by a family of [[seminorm]]s {''p''<sub>&alpha;</sub>|&alpha;&isin;''A''} where ''A'' is an index set. Let ''M'' be a closed subspace, and define seminorms ''q''<sub>&alpha</sub> by on ''X''/''M''
METRIZABLE ..
 
..
El quocient d'un [[espai localment convex]] per un subespai tancat és una altra vegada localment convex {{harv|Dieudonné|1970|loc=12.14.8}}. En efecte, suposi que ''X'' és localment convex de manera que la topologia en ''X'' és generat per una família de [[norma (matemàtiques)|seminormes]] {''pàg.'' <sub>α;</sub>|α;∈''A'' } on ''A'' és un conjunt d'índex. Sia ''M'' un subespai tancat, i definir seminormes ''q'' <sub>α</sub> per en ''X'' /''M''
FRÉCHET SPACE ..
 
..
 
QUOTIENT SET ..
 
..
:<math>q_\alpha([x]) = \inf_{x\in [x]} p_\alpha(x).</math>
QUOTIENT GROUP ..
 
..
 
QUOTIENT MODULE ..
 
..
Then ''X''/''M'' is a locally convex space, and the topology on it is the [[quotient topology]].
QUOTIENT SPACE ..
 
..
Llavors ''X'' /''M'' és un espai localment convex, i la topologia en això és la [[topologia de quocient]].
TOPOLOGY ..
 
 
 
If, furthermore, ''X'' is [[metrizable]], then so is ''X''/''M''. If ''X'' is a [[Fréchet space]], then so is ''X''/''M'' {{harv|Dieudonné|1970|loc=12.11.3}}.
 
Si, a més ''X'' és [[metrizable]], llavors així és ''X'' /''M'' . Si ''X'' és un [[Espai de]] FRÉCHET, llavors així és ''X'' /''M'' {{harv|Dieudonné|1970|loc=12.11.3}}.
 
 
 
==See also==
 
== Vegeu també ==
*[[quotient set]]
 
* [[el quocient]] S'ENDURIA
*[[quotient group]]
 
* [[grup quocient]]
*[[quotient module]]
 
* [[mòdul de]] QUOCIENT
*[[quotient space]] (in [[topology]])
 
* [[espai quocient]] (en la [[topologia|topologia)]]
 
 
 
==References==
 
== Referències ==
* {{citation|first=Paul|last=Halmos|authorlink=Paul Halmos|title=Finite dimensional vector spaces|publisher=Springer|year=1974|isbn=978-0387900933}}.
 
* {{Ref-llibre|first = Paul|last = Halmos|authorlink = Paul Halmos|títol = Finite dimensional vector spaces|editorial = Springer|any = 1974|isbn = 978-0387900933}}.
* {{citation|first=Jean|last=Dieudonné|authorlink=Jean Dieudonné|title=Treatise on analysis, Volume II|publisher=Academic Press|year=1970}}.
 
* {{Ref-llibre|first = Jean|last = Dieudonné|authorlink = Jean Dieudonné|títol = Treatise on analysis, Volume II|editorial = Academic Press|any = 1970}}.
 
 
 
[[Category:Linear algebra]]
[[Category:Functional analysis]]
 
[[de:Faktorraum]]
[[it:Spazio vettoriale quoziente]]
[[he:מרחב מנה (אלגברה לינארית)]]
[[pl:Przestrzeń ilorazowa (algebra liniowa)]]
[[ru:Факторпространство по подпространству]]
[[zh:商空间 (线性代数)]]
[[en:Quotient space (linear algebra)]]
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