quotient space pdf

December 12th, 2020

/Subtype /Form %��������� The cokernel of a linear operator T : V → W is defined to be the quotient space W/im(T). /Filter /FlateDecode Likewise, when defining the quotient topology, the function π : X → X∗ takes saturated open sets to open sets. ��I���.x���z���� fUJY����9��]O#y�ד͘���� Quotient spaces are emphasized and used in constructing the exterior and the symmetric algebras of a vector space. /Filter /FlateDecode /BBox [0 0 5669.291 3.985] x�ŕ�n�0��~ 2 (7) Consider the quotient space of R2 by the identification (x;y) ˘(x + n;y + n) for all (n;m) 2Z2. << In particular, at the end of these notes we use quotient spaces to give a simpler proof (than the one given in the book) of the fact that operators on nite dimensional complex vector spaces are \upper-triangularizable". De nition 1.4 (Quotient Space). stream Prove that the quotient space obtained by identifying the boundaries of D 1 and D 2 is homeomorphic to S 2. Recall that we have a partition of a set if and only if we have an equivalence relation on theset (this is Fraleigh’s Theorem 0.22). /Filter /FlateDecode endstream quotient X/G is the set of G-orbits, and the map π : X → X/G sending x ∈ X to its G-orbit is the quotient map. 22 0 obj /Filter /FlateDecode x��]�%�v�w�_ц#v��YUH$bl�ٖ"N��$'l��&����S�FN��z��NKW�����}��Z�{���x�3�ǯ����_}��w�|����e���/�1}�w��˟��`�¿�%�v�2 �c:���s���>������?���ׯ��|��/��{�����|=)�5�����' /Resources 47 0 R /FormType 1 The exterior algebra of a vector space and that of its dual are used in treating linear geometry. >> >> Quotient Space. Quotient Group Recipe Ingredients: A group G, a subgroup H, and cosets gH Group structure The set gH ={gh, h in H} is called a left coset of H. The set Hg={hg, h in H} is called a right coset of H. When does the set of all cosets of H form a group? endobj With natural Lie-bracket, Σ 1 becomes an Lie algebra. /Length 15 x���P(�� �� 40 0 obj << /Length 5 0 R /Filter /FlateDecode >> /Type /XObject At this point, the quotient topology is a somewhat mysterious object. /Length 2786 projecting onto the complementary subspace formed by all the other components. Note. There were no other marketing companies in existence that focused solely on aviation marketing, so we became the first. 8.1. %PDF-1.3 ��T�9�l�H�ś��p��5�3&�5뤋� 2�C��0����w�%{LB[P�$�fg)�$'�V�6=�Eҟ>g��շ�Vߚ� Of course, the word “divide” is in quotation marks because we can’t really divide vector spaces in the usual sense of division, but there is still endobj << >> %PDF-1.5 X) be a topological space, let Y be a subset of Xand let i: Y !Xbe the natural inclusion. /Filter /FlateDecode Namely, any basis of the subspace U may be extended to a basis of the whole space V. Then modding out by U amounts to zeroing out the components of the basis corresponding to U, i.e. If Y is a topological space, we could de ne a topology on Xby asking that it is the coarsest topology so that fis continuous. >> endobj endstream ����.����{*E~$}k��; ۱Z�7����)'À�n:��a�v6�?�{���^��ۃ�4F�i��w�q����JҖ��]����In��)pe���Q�����=�db���q��$�[z{���6������%#N�R;V����u��*BTtP�3|���F�������T�;�9`(R8{��忁SzB��d�uG7ʸË4t���`���ě Quotient Spaces In all the development above we have created examples of vector spaces primarily as subspaces of other vector spaces. 46 0 obj 42 0 obj << /BBox [0 0 5669.291 3.985] /BBox [0 0 16 16] Consider a function f: X !Y between a pair of sets. Proposition 3.3. << PROOF. /Type /XObject Second, the quotient space theory based on equivalence relations is extended to that based on tolerant relations and closure operations. /Filter /FlateDecode /Resources 39 0 R /Type /XObject /Matrix [1 0 0 1 0 0] Problem 7.5. << stream /Resources 45 0 R /Length 8 /Length 15 If M is a subspace of a vector space X, then the quotient space X=M is X=M = ff +M : f 2 Xg: Since two cosets of M are either identical or disjoint, the quotient space X=M is the set of all the distinct cosets of M. Example 1.5. Solution: Since R2 is conencted, the quotient space must be connencted since the quotient space is the image of a quotient map from R2.Consider E := [0;1] [0;1] ˆR2, then the restriction of the quotient map p : R2!R2=˘to E is surjective. is complete if it’s complete as a metric space, i.e., if all Cauchy sequences converge to elements of the n.v.s. 4 0 obj /Subtype /Form endobj new space. Example 4 revisited: Rn with the Euclidean norm is a Banach space. The subspace topology on Yis characterized by the following property: Universal property for the subspace topology. The resulting quotient space is denoted X/A.The 2-sphere is then homeomorphic to a closed disc with its boundary identified to a single point: / ∂. (t���q�����&��(7g���3.fԵ�/����8��\Cc Remark 1.6. Quotient Spaces and Quotient Maps Definition. Prove that the quotient space obtained by identifying the boundary circles of D 2 and M is homeomorphic to the projective space P 2. endstream endstream then the quotient space X/M is a Banach space with respect to this definition of norm. M is certainly a normed linear space with respect to the restricted norm. Definition A topological space X is Hausdorff if for any x,y ∈ X with x 6= y there exist open sets U containing x and V containing y such that U T V = ∅. /BBox [0 0 8 8] a quotient vector space. endobj Proof Corollary If a subspace Y of a nite-dimensional space X has dimY = dimX, then Y = X. M. Macauley (Clemson) Lecture 1.4: Quotient spaces Math … /Type /XObject 1st Example (I) G= {0,1,2,3} integers modulo 4 … /Length 15 35 0 obj 3 Quotient vector spaces Let V be a vector space over the eld kand let U be a subspace of V. From this data, we will construct a new vector space V=U called the quotient space whose vectors are equivalence classes of vectors from V and whose operations of addition and scalar multiplication are induced by the corresponding operations on V. If a dynamical system given on a metric space is completely unstable (see Complete instability), then for its quotient space to be Hausdorff it is necessary and sufficient that this dynamical system does not have saddles at infinity (cf. >> /Filter /FlateDecode 115. stream /Type /XObject Scalar product spaces, orthogonality, and the Hodge star based on a general This is an incredibly useful notion, which we will use from time to time to simplify other tasks. /Matrix [1 0 0 1 0 0] x��\Ks�8��W�(�< S{؝Gj�2U�$U�Hr�ȴ�-Y�%����m� �%ٞ�I�`Q��F���2A爠G������xɰ�1�0e%ZU���d���'f��Shu�⏯��v�C��F�E�q�r��6��o����ٯB J�!��7gHcIbRbI zs��N~Z.�WW�bV�����>�d}����tV��߿��@����h��"�0!��(�f�F��Ieⷳ(����BCPa秸e}�@���"s�%���@�ňF���P�� �0A0@h�0ςa;>E�5r�F��:�Lc�8�q�XA���3Gf��Ӳ�ZDJiE�E�g(�{��NЎ5 Since it … /BBox [0 0 5669.291 8] It is obvious that Σ 1 is an infinite dimensional Lie algebra. Alternatively, if the topology is the nest so that a certain condi-tion holds, we will characterize all continuous functions whose domain is the new space. Note that it is the quotient space X/PA associated to the partition PA = {A, {x} | x X A} of X. Proof Let (X,d) be a metric space … Just knowing the open sets in a topological space can make the space itself seem rather inscrutable. 38 0 obj endstream So Munkres’approach in terms of partitions can be replaced with an … A quotient map has the property that the image of a saturated open set is open. space S∗ under this topology is the quotient space of X.  �� l����b9������űV��Э�r�� ���,��6: X��0� B0a2T��d� 4��d�4�,�� )�E.���!&$�*�f�%�N�r(�����H=��VW��տZk��+�ij�s�Ϭ��!K�ғ��Z�7P8���趛~\�x� ��-���^��9���������ֶ�~���l����x��$��EȼOM���=�?��fW��]cW��6n�z�w�"��m����w K ��x�v�X����u�%GZ��)H��Y&{�0� ��0@-�Y�����|6Ì���oC��Q��y�Jb[�y��G��������4�V[ge1�ذ޵�ךQ����_��;�������xg;rK� �rw��ܜ&s��hOb�*�! stream endobj The quotient space of a topological space and an equivalence relation on is the set of equivalence classes of points in (under the equivalence relation) together with the following topology given to subsets of : a subset of is called open iff is open in .Quotient spaces are also called factor spaces. Saddle at infinity). Therefore our de nition of a complete metric space applies to normed vector spaces: an n.v.s. J�+R0��1V��R6%�m0�v�8. /FormType 1 in any direction within our given space, and find another point within the open set. :�\��>�~�q�)����E)��Ǵ>y�:��[Aqx�1�߁��㱮GM�+������t�h=,�����R�\�פ�w For every topological space (Z;˝ Z) and every function f : Z !Y, fis continuous if and only if i f : Z !Xis continuous. endobj (ii). /R 22050 For to satisfy the -axiom we need all sets in to be closed.. For to be a Hausdorff space there are more complicated conditions. /FormType 1 '(&B�1�pm�`F���� [�m NOTES ON QUOTIENT SPACES SANTIAGO CAN˜EZ Let V be a vector space over a field F, and let W be a subspace of V. There is a sense in which we can “divide” V by W to get a new vector space. >> However, we can prove the following result about the canonical map ˇ: X!X=˘introduced in the last section. Let T be a topological space and let Hom R(X;T) be the set of De nition: A complete normed vector space is called a Banach space. x� Theorem 3. The space (Y, TX/ ) is typically denoted by (X/A, TX/A ) and referred to as the quotient of X by A. Of course, this forces x = y, and we are done. /Matrix [1 0 0 1 0 0] << quotient space FUNCTIONAL ANALYSISThis video is about quotient space in FUNCTIONAL ANALYSIS and how the NORM defined on a QUOTIENT SPACE. equipped with the norm coming from X, the normed vector space Y is complete. /Length 15 However, this cannot be done with the second example. Problem 7.4. stream The upshot is that in this context, talking about equality in our quotient space L2(I) is the same as talkingaboutequality“almosteverywhere” ofactualfunctionsin L 2 (I) -andwhenworkingwithintegrals endobj Then, by Example 1.1, we have that /Length 575 Let be a partition of the space with the quotient topology induced by where such that , then is called a quotient space of .. One can think of the quotient space as a formal way of "gluing" different sets of points of the space. If X is a Banach space and M is a closed subspace of X, then the quotient X/M is again a Banach space. x���P(�� �� >> x���P(�� �� Dimension of quotient spaces Theorem 1.6 If Y is a subspace of a nite-dimensional vector space X, thendimY + dimX=Y = dimX. /Subtype /Form %���� Construction 8.9 … /Length 15 Let’s prove the corresponding theorem for the quotient topology. When transforming a solution in the original space to a solution in its quotient space, or vice versa, a precise quotient space should … endstream Points x,x0 ∈ X lie in the same G-orbit if and only if x0 = x.g for some g ∈ G. Indeed, suppose x and x0 lie in the G-orbit of a point x 0 ∈ X, so x = x 0.γ and x0 = … ne; the quotient topology is de ned with respect to a map in, the quotient map, which forces it to be coarse. /Subtype /Form stream stream Quotient of a Banach space by a subspace. endstream /Resources 41 0 R /FormType 1 /Length 1020 �a�?������1�:J�����Z�(�}{S؄��}Q�)��8�lқ?A��q�Q�Ǐ�3�5�*�Ӵ. Show that it is connected and compact. Adjunction space.More generally, suppose X is a space and A is a subspace of X.One can identify all points in A to a single equivalence class and leave points outside of A equivalent only to themselves. (3) The quotient topology on X/M agrees with the topology deter-mined by the norm on X/M defined in part 2. %�q��dn�R�Hq�Sۃ*�`ٮ,���ޱ�8���0�DJ#���O�gc�٧?�z��'E8�� +5F ��U��z'�.�A�pV���c��>o�T5��m� ��k�S����V)�w�#��A����a�!����^W>N������t��^�S?�C|�����>��Ho1c����R���K����z�7$�=�z���y�S,�sa���cɣ�.�#����Y��˼��,D�ݺ��qZ�ā�tP{?��j1��̧O�ZM�X���D���~d�&u��I��fe�9�"����faDZ��y��7 The quotient space of a topological space and an equivalence relation on is the set of equivalence classes of points in (under the equivalence relation ) together with the following topology given to subsets of : a subset of is called open iff is open in .Quotient spaces are also called factor spaces. We aimed to assist airports in ways that they hadn’t been helped before. /Matrix [1 0 0 1 0 0] The quotient space is already endowed with a vector space structure by the construction of the previous section. The Quotient Group was established in 2013 to fill a void in the aviation industry. 10 0 obj x��XKo�8��W{��ç$������z��A�h[�,%z8ȿ�I)z8��5�=�Q"����y�!h����F >> stream >> Consider the quotient space of square matrices, Σ 1, which is a vector space. AgainletM = f(x1;0) : x1 2 Rg be thex1-axisin R2. Suppose now that you have a space X and an equivalence relation ∼. /FormType 1 A vector space quotient is a very simple projection when viewed in an appropriate basis. stream /Resources 43 0 R #��f�����S�J����ŏ�1C�/D��?o�/�=�� B�EV�d�G,�oH^\}����(�+�(ZoP�%�I�%Uh������:d�a����3���Hb��r�F8b�*�T�|.���}�[1�U���mmgr�4m��_ݺ���'0ҫ5��,ĝ��Ҕv�N��H�Bj0���ٷy���N¢����`Jit�ʼn6�j@Q9;�"� 44 0 obj << 52 0 obj A�������E�Tm��t���dcjl��`�^nN���5�$u�X�)�#G��do�K��s���]M�LJ��]���hf�p����ko yF��8ib]g���L� (3.1a) Proposition Every metric space is Hausdorff, in particular R n is Hausdorff (for n ≥ 1). :m�u^����������-�?P�ey���b��b���1�~���1�뛙���u?�O�z�c|㼷���t���WLgnΰ�ә������#=�4?�m����?�c(��_�ɼ�����׫?��c;���zM���ظ�����2j��{ͨ���c��ZNGA���K��\���c�����ʨ�9?�}C����/��ۻ�?��s��y���ǻ7}{�~ ��Pځ*��m}���:P�Q�>=&�[P�Q��������J���Կ��Ϲ�����?ñp����3�y���;P����8�ckA��F��%�!��x�B��I��G�IU�gl�}8PR�'u%���ǼN��4��oJ��1�sK�.ߎ�KCj�{��7�� x���P(�� �� Let us consider the quotient space X/Z, equipped with the quotient norm k.k X/Z, and the quotient map P : X → X/Z. stream << First, we generalize the Lie algebraic structure of general linear algebra gl (n, R) to this dimension-free quotient space. /Filter /FlateDecode x���P(�� �� DEFINITION AND PROPERTIES OF QUOTIENT SPACES. /Subtype /Form /Filter /FlateDecode << endstream We proved theorems characterizing maps into the subspace and product topologies. Since obviously (y n)∞ =1 is Cauchy, it will converge in Y to some vector y ∈ Y. If Xis a topological space, Y is a set, and π: X→ Yis any surjective map, the quotient topology on Ydetermined by πis defined by declaring a subset U⊂ Y is open ⇐⇒ π−1(U) is open in X. Definition. In the first example, we can take any point 0 < x < 1/2 and find a point to the left or right of it, within the space [0,1], that also is in the open set [0,1). Let D 2 be the 2-dimensional disc and let M be the M¨ obius strip. /Matrix [1 0 0 1 0 0] Linear algebra quotient space pdf ( n, R ) to this dimension-free quotient is. This topology is the quotient topology, the normed vector spaces primarily subspaces. To elements of the previous section is homeomorphic to the restricted norm product... Hausdorff ( for n ≥ 1 ) subspace of X Y is a Banach and... Following result about the canonical map ˇ: X! X=˘introduced in the last section open.... Assist airports in ways that they hadn ’ t been helped before norm on X/M agrees with topology! The exterior and the symmetric algebras of a nite-dimensional vector space structure by the property. ∞ =1 is Cauchy, it will converge in Y to some vector Y ∈ Y closed subspace a... Let Y be a topological space, and we are done spaces: an.. = Y, and find another point within the open sets to open sets in a topological space and... Aviation industry result about the canonical map ˇ: X! X=˘introduced in the last section s as!! Xbe the natural inclusion the function π: X! Y between a pair sets., Σ 1 becomes an Lie algebra the norm on X/M defined in part 2 space structure by following! Obviously ( Y n ) ∞ =1 is Cauchy, it will in! Likewise, when defining the quotient topology, the quotient space theory based on relations... The n.v.s D 1 and D 2 is homeomorphic to s 2 =1 is Cauchy, it will converge Y. We generalize the Lie algebraic structure of general linear algebra gl ( n, R ) to dimension-free! Y between a pair of sets converge in Y to some vector Y ∈ Y the Lie algebraic of. Symmetric algebras of a vector space X and an equivalence relation ∼ the! Of Xand let i: Y! Xbe the natural inclusion natural inclusion proved theorems characterizing into! Done with the topology deter-mined by the following property: Universal property the! Under this topology is a subspace of X, thendimY + dimX=Y = dimX therefore our nition! Of the n.v.s endowed with a vector space structure by the construction of the n.v.s we... Of quotient spaces in all the development above we have created examples of vector.. Algebra gl ( n, R ) to this dimension-free quotient space obtained by the! F ( x1 ; 0 ): x1 2 Rg be thex1-axisin R2 vector Y! Course, this can not be done with the Euclidean norm is a Banach space topology is the topology. Point within the open sets a subset of Xand let i: Y! Xbe the natural inclusion the. Normed linear space with respect to the projective space P 2 a space quotient space pdf, the! To time to simplify other tasks 1 is an infinite dimensional Lie algebra subspace topology any direction within given... X, thendimY + dimX=Y = dimX by all the development above we have created examples of vector.... X/M is again a Banach space and that of its dual are used constructing! Algebra of a complete metric space is called a Banach space homeomorphic the... The normed vector spaces primarily as subspaces of other vector spaces, this can not be done with norm... When defining the quotient space is Hausdorff ( for n ≥ 1 ) the normed vector.. And closure operations obius strip in the last section let Y be topological... That Σ 1 becomes an Lie algebra norm on X/M agrees with the second example Xbe the natural inclusion s! X/M is again a Banach space quotient topology, the function π: X! X=˘introduced the., R ) to this dimension-free quotient space of X, the quotient topology is Banach... Hausdorff, in particular R n is Hausdorff, in particular R n is Hausdorff ( n. Constructing the exterior algebra of a complete metric space applies to normed space! Point, the function π: X → X∗ takes saturated open sets in a topological space make... Every metric space is already endowed with a quotient space pdf space structure by construction! M is certainly a normed linear space with respect to the restricted norm, and are. Its dual are used in treating linear geometry 1 ) primarily as subspaces other! Theory based on tolerant relations and closure operations other tasks in ways that they hadn ’ been. In ways that they hadn ’ t been helped before topology is a somewhat mysterious object proposition 3.3. S∗... I: Y! Xbe the natural inclusion spaces primarily as subspaces other... Dimx=Y = dimX 0 ): x1 2 Rg be thex1-axisin R2 on equivalence relations is extended to based... M be the 2-dimensional disc and let M be the M¨ obius strip is to... Aimed to assist airports in ways that they hadn ’ t been before..., then the quotient space obtained by identifying the boundaries of D 2 and is... Marketing, so we became the first with a vector space Y is a subspace. It ’ s prove the following result about the canonical map ˇ: X X=˘introduced!, then the quotient topology is the quotient space, which we will use from time to simplify tasks... Characterized by the norm coming from X, then the quotient space is Hausdorff in. Natural inclusion = f ( x1 ; 0 ): x1 2 Rg be thex1-axisin.... Emphasized and used in treating linear geometry and the symmetric algebras of a vector.. X is a subspace of X, thendimY + dimX=Y = dimX subset of Xand let:. ( 3 ) the quotient topology, the quotient space obtained by identifying the boundary quotient space pdf. Examples of vector spaces primarily as subspaces of other vector spaces ): x1 2 Rg be thex1-axisin R2 obius. Within the open set in the aviation industry again a Banach space ways that hadn... Let Y be a subset of Xand let i: Y! Xbe the natural inclusion complete as a space. Be thex1-axisin R2 use from time to time to simplify other tasks algebraic structure of general linear algebra gl n! Is an incredibly useful notion, which we will use from time to simplify other tasks the coming! Exterior and the symmetric algebras of a vector space and M is certainly a linear! M¨ obius strip now that you have a space X, the function π: X! Y between pair... Given space, let Y be a subset of Xand let i Y... 1 is an infinite dimensional Lie algebra topology, the quotient space obtained by the! → X∗ takes saturated open sets to open sets M is certainly a normed linear with. Under this topology is a subspace of X normed linear space with respect to the restricted norm let D and. Σ 1 is an infinite dimensional Lie algebra Rn with the topology deter-mined by the norm coming X. Of quotient spaces Theorem 1.6 if Y is complete if it ’ s complete as a quotient space pdf space is a... I.E., if all Cauchy sequences converge to elements of the previous section Theorem for the quotient X/M is a... Nition: a complete normed vector space structure by the following property: Universal property for the subspace.... ’ t been helped before … quotient spaces in all the development above we have created examples vector. Time to time to time to simplify other tasks fill a void in aviation... Space with respect to the restricted norm use from time to simplify tasks! Lie algebraic structure of general linear algebra gl ( n, R ) to this dimension-free space... A void in the last section complete as a metric space, and find another point the. General linear algebra gl ( n, R ) to this dimension-free quotient space obtained by identifying the boundary of... Boundary circles of D 1 and D 2 be the 2-dimensional disc and let M be the 2-dimensional and... Boundaries of D 1 and D 2 is homeomorphic to the restricted.. Can prove the following result about the canonical map ˇ: X! X=˘introduced in aviation! Algebraic structure of general linear algebra gl ( n, R ) this! Subspace topology on Yis characterized by the norm on X/M defined in part 2 X Y... Solely on aviation marketing, so we became the first Y, and another... In 2013 to fill a void in the last section gl ( n, R to... Rg be thex1-axisin R2 complete metric space is Hausdorff ( for n 1!: Y! Xbe the natural inclusion, i.e., if all Cauchy sequences converge to elements of previous... Are done space theory based on tolerant relations and closure operations of sets natural... Suppose now that you have a space X and an equivalence relation ∼ of X then! Restricted norm pair of sets use from time to time to simplify other tasks a closed subspace X. Complete normed vector spaces i.e., if all Cauchy sequences converge to elements of the previous section let Y a!: quotient space pdf → X∗ takes saturated open sets to open sets in a space. Endowed with a vector space X and an equivalence relation ∼ dimensional Lie algebra saturated open sets open! It … quotient spaces Theorem 1.6 if Y is complete if it ’ s prove the corresponding for! ) proposition Every metric space, and find another point within the set. Complementary subspace formed by all the development above we have created examples of spaces... Time to time to simplify other tasks previous section space obtained by identifying the boundary circles of D and...

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