metric space examples proof

We do not develop their theory in detail, and we leave the veriﬁcations and proofs as an exercise. stream [���%�xjR�JM�S3Uq��n�QK-�������H�N;s�H������7�)�H�e�'�WL�L��Hi5��O~I��k!������O�^���{�'8E:���t2%��y�~�or׍�(F� �m�=�F҈^�xw1%R%S�Ɔ�I�Z�����)F�J��bHR:i��+,Y���T�`L[��4ǄU��)�4�V��,�F���T! /Length 3871 x��\I��6��W���. x��َ���}�B�'�ðor~ȱIl� � �~�J��)��������橖4cO�\$/R���uuUu1Y�-�ş�������ퟘY0���v���ǌ��I�8�lq�Z|��jms}#�������m],��~�/����o�Z�\$Β�!�&D��lq�U,DF�n7���7\��\$�\Ȩ(�y�uU�KK��Ə]V���[�Tk�����xY���g������r��f�x�/��lh��ęJ���a������6���b���?�����%5ڦ�t�"���,*��n��p��-���р#�Ȋ��u�Mh�Lé5b�y�A\�� Examples of metric spaces. 3 0 obj << 5 0 obj << The following properties of a metric space are equivalent: Proof. /Filter /FlateDecode The fact that every pair is "spread out" is why this metric is called discrete. x��ZKs7��WpO�M������M\vT�j�*�aL��)��"ɿ~�ј�@Y^˱{���F�я�{��ӣo��� ��0'��*��g���o�/'�O_�ڻ�x�dv�+v|�������'��F��4q�e��?����~���cag�yk5�e5�n���9�h�''/~��'��2;=y���I��r��*? /Filter /FlateDecode Assume therefore that x 6= 0 and y 6= 0. In most cases, the proofs �`P���i�w?�[����>rbWg�Y�vhl7��n*��O�K:}��vR�!�9#�]������l��d�i�PN��VpV�#uDp��ݳ�6|]�M�[��K�A1���J(�F�q@ H ��!/�T-c�SZ�\$����ZKr��Z� �|.����ĭ�?�����F��b��/���\$��h���m�WE���/AI��E�� |i S���b�f]�MHgA)��9V �q� �j�Bӆ�^�����iY��V ��)�Қ�c��E�d{|l8���Tx� ��ȼ�,�i Y!��~ź��yg��L��P�CX���cU2{9 e����F�e&m�3J�1������8߭K̥|N�����d5��H���h�{�T�CY ��K꼩���2N����[³�����SƍU�gL=o�Wh�m� �S���&o����� 2. Let C[0, 1] be the set of all continuous R-valued functions on the interval [0, 1]. stream %���� The set of real numbers R with the function d(x;y) = jx yjis a metric space. 1. �f���~��=��p�˰�(��ƽ׳�G�:����\$������G�9q�6F� �Hu�@��[�^�/d�;P-��Ğ [V�; 8\$G'T���EI���`��R) �~����.9yHr�S ɩ��侻��B|��+J�q��Xsn��x��v�݊>��1��k��ў�M��ܠ���� *{PS���_Ӏ}H_�J.��iC),�� c���H�Y!a This space (X;d) is called a discrete metric space. %PDF-1.5 Metric Spaces A metric space is a set X that has a notion of the distance d(x,y) between every pair of points x,y ∈ X. Assume that is not sequentially compact. Euclidean Space and Metric Spaces 8.1 Structures on Euclidean Space ... EUCLIDEAN SPACE AND METRIC SPACES Examples 8.1.2. 20 0 obj << is /Filter /FlateDecode 1.1. Theorem. Any incomplete space. >> Any unbounded subset of any metric space. )O"�cd%Q���D��Z�Hdz³. %PDF-1.5 ��]�3�G)b�q;�S��R����2}bl~������AK�:�`~M�M0��U]4U}v�#ثA�h@B�˼�DХj�����l�1+��u�1�Yݝ�*��u�T�;�S�C�QP �k���=Y�]T� ����e���2'��(�ϙ�����q� Then ε = 1 2d(x,y) is positive, so there exist integers N1,N2 such that d(x n,x)< ε for all n ≥ N1, d(x n,y)< ε for all n ≥ N2. Therefore our de nition of a complete metric space applies to normed vector spaces: an n.v.s. Proof. stream ���ot&����C@�!��.om����aU:@^�v����Mh��M���Yd�W7�a+�*���UPxh���K=r�!o���O-��R�;�1�yq�Ct5m^��u]���,��h�H��޷��_��Y�| �vEӈ��M�ԭhC�[Vum��ܩ�UQށX ��` �':v�udPۺ���ӟ�4���#5�� �(,""M��6�.z͢��x��d��}�v�obwL��L��Yo������+�S���o����Ǐ��� 3. is complete and totally bounded. /Length 2734 We now give examples of metric spaces. [΄�L %���� �r��a�6�=���r�H�&�� ����n]8�Rڙ�ҏn]D�([�)���l�O� ��DL�� �Q_Gm�%Ǝ^�P���3��Fŕ;������^ 9�b��]���!i��������E����V�\���������J�&��(a�Rr�QY''!1:eۣ��doʆɡ�H�yZ����Zɔ�_��8�F~p�J�@o��@z���@�T���V�����)*A���%&�m�ᐭ��]h�:8���Vؘ LN��ϰ��@)x ¥|�K��I*����u�.!���o�fN����Jg�����J�**h���:%Li]�ۇ�! F+��G1+9�yQ6�j �s����m�s)�eY�w!h��Ex�����r��Fdg��z.��\��e�y��ZWm� �f����V�%�YM�hZ��ۺ��e�A�;Xǁ�fY�����ž.���i�����-�����*۞ѓ�Rޭ�MIc�U�ZUSS㢾�e)��kCi&��Hf�l�W0���:��5,E��5�v��\$ �xn�%������ "'F�9��,�=`/��Ԡb��o����蓇�. Proof. Discrete metric space is often used as (extremely useful) counterexamples to illustrate certain concepts. Example 1.1. Non-examples. Recall that every normed vector space is a metric space, with the metric d(x;x0) = kx x0k. 1. /Length 3785 Example 1. �%��)�V1�����hj�J3�c��? >�h_�Pc4Ȏw~㲤J�������V�yG ��&��Hft�(Y���)����?�MCc]�Oz+`h �@�r��߄���J����>�����Hjp��ai����.��I�^�t��=yƸ���=t�}ý��jq��:��Ş�(ޅ�0)̗�� `3b�)���^�z]�&Ve�,� In most of the examples the conditions (1) and (2) of De nition 1.1 are easy to verify, so we mention these conditions only if there is some di culty in establishing them. Proof: Exercise. Show that the real line is a metric space.