complete metric space examples pdf

The purpose of this chapter is to introduce metric spaces and give some deﬁnitions and examples. Normed real vector spaces) 95 0 obj endobj (4.1. Properties of complete spaces) 11 0 obj endobj xڕW�r�8��+X��^1;w��x*gWMW�̂`�V��X��W��0��Q6��=瞫��n��;��˫O78� �(��C � !�@��-W��l�8�� z�ž,��zzg��׳��˿F@�n�X�P�+J ��$d�Y��$��7�6 0��{��6 �!�kϾ<6F{�H�A�x�!��5�I �\�B��L%OHz?1�>>d~��5�Z�_�.Df�wi��)0��}��L��`�C\��{��휹WE�W��־x�U��3�A?q�\}��]�͈� ��5��q��-��\��ؘ�@��e��,�!�r��뀍�SJ v�˲�F�@�4ϑO��T61K��Y��}�S��7\��D!L*%��ĝ��Jx��2�Cğe��0��ԥpC3�#W�H��Te��R��&��_��ufѝ�?U�U��F��A��=��1�y��U��끍z%��r�G��I-9Ɲ��&��\ނ�-sdK�>�z�9aJ�O�3:�B��&�߀ � B5�� Nec{�:j�п��w�:�U�'J)^L%o]��E�M��뻶�_��"W�X/�gj_� 119 0 obj 64 0 obj 28 0 obj Continuity improved: uniform continuity) endobj 4 0 obj Then (1 n) is a Cauchy sequence which is not convergent in X. Deﬁnition 3. (3.1. 39 0 obj endobj (=)) Let x2S. 8 0 obj endobj 1.5 Theorem. << /S /GoTo /D (subsection.12.2) >> 52 0 obj (12.1. << /S /GoTo /D (subsection.2.2) >> 116 0 obj 1. 91 0 obj endobj 88 0 obj endobj endobj Thus, fx ngconverges in R (i.e., to an element of R). endobj 48 0 obj 115 0 obj endobj (6. endobj (3. %PDF-1.4 15 0 obj endobj endobj endobj 100 0 obj endobj 96 0 obj �8Ik��ÄIV�Z��Ӻ�vj��"k��R�1c��Ӡ�4��A�E�aC��:��|1��kk��Л��?�LI��;l|S��r��C\`�L,c��k�tu��d�ν�в{�X��ot�$��&�h��I�e�m6�M�Z��}�4[��C�qU*r��o!��N�vV�l]��. Convergence of sequences in metric spaces) 36 0 obj endobj Set theory revisited70 11. 24 0 obj Obviously, this sequence is a Cauchy sequence, and, since Sis complete, it converges to some x~ 2S. endobj endobj << /S /GoTo /D (subsection.8.2) >> endobj /Length 944 (2.2. endobj 7 0 obj (8.2. 3 0 obj 112 0 obj Product spaces) endobj The completion of a metric space) >> endobj One measures distance on the line R by: The distance from a to b is |a - b|. 87 0 obj Let (X;d) be a complete metric space and S X. 27 0 obj (9. Path-connected spaces) << /S /GoTo /D (subsection.6.1) >> Compact spaces) Examples. 16 0 obj endobj 51 0 obj endobj is a complete metric space iff is closed in Proof. Examples include the real numbers with the usual metric, the complex numbers, finite-dimensional real and complex vector spaces, the space of square-integrable functions on the unit interval L^2([0,1]), and the p-adic numbers. << /S /GoTo /D (section.5) >> Topological spaces) endobj 80 0 obj endobj 55 0 obj (8. 35 0 obj endobj Note that each x n is an irrational number (i.e., x n 2Qc) and that fx ngconverges to 0. 75 0 obj 67 0 obj endobj Properties of open subsets and a bit of set theory) Assume that is closed in Let be a Cauchy sequence, Since is complete, But is closed, so On the other hand, let be complete, and let be a limit point of so (in ), . 47 0 obj endobj (6.2. Let S be a closed subspace of a complete metric space X. Theorem 4. Dealing with topological spaces) endobj A very basic metric-topological dictionary) endobj Interlude II) endobj endobj (10. (11.2. (3.2. 76 0 obj endobj << /S /GoTo /D (subsection.3.1) >> 19 0 obj 108 0 obj 40 0 obj endobj For example if I change real numbers into rational number with usual metric ( absolute value ) it would be incomplete. << /S /GoTo /D (section.2) >> endobj << /S /GoTo /D (section.4) >> 71 0 obj << /S /GoTo /D (subsection.12.1) >> �Q4b�u�a��'0U7?�OϤ�H�$6E�BG endobj 63 0 obj On the other hand if have a some kind of metric on some space it would be incomplete though. Proof. Quotient topological spaces) 44 0 obj 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. Equivalent metrics) /Filter /FlateDecode (10.1. (5. << /S /GoTo /D (section*.1) >> Complete spaces) What topological spaces can do that metric spaces cannot82 12.1. 31 0 obj 43 0 obj (6.1. A complete metric space is a metric space in which every Cauchy sequence is convergent. If is the real line with usual metric, , then Remarks. 23 0 obj 56 0 obj Proof. From metric spaces to topological spaces75 11.2. endobj endobj We do not develop their theory in detail, and we leave the veriﬁcations and proofs as an exercise. << /S /GoTo /D (subsection.11.1) >> (4. Homeomorphisms of metric spaces and open maps) Continuous functions between metric spaces) 84 0 obj endobj endobj << /S /GoTo /D (subsection.4.1) >> One-point compactification of topological spaces) A very basic metric-topological dictionary78 12. (11.1. 68 0 obj << /S /GoTo /D (section.6) >> 120 0 obj A closed subset of a complete metric space is a complete sub- space. endobj Completion of a metric space A metric space need not be complete. endobj 60 0 obj << /S /GoTo /D (section.9) >> Complete spaces54 8.1. Metric spaces: basic definitions) endobj 99 0 obj << /S /GoTo /D (subsection.2.1) >> (REFERENCES) 2. << /S /GoTo /D (subsection.10.1) >> << /S /GoTo /D [121 0 R /Fit ] >> To make space incomplete either i can change the metric or the ambient space. endobj Proposition 1.1. Dealing with topological spaces72 11.1. (12.2. Properties of complete spaces58 8.2. << /S /GoTo /D (subsection.8.1) >> (1. (2. 152 0 obj << Interlude II66 10. (11. endobj %�� endobj Then Sis completeifandonlyifSisclosed. << /S /GoTo /D (section.8) >> Definition and examples of metric spaces. << /S /GoTo /D (subsection.3.2) >> The completion of a metric space61 9. << /S /GoTo /D (subsection.3.3) >> endobj endobj << /pgfprgb [/Pattern /DeviceRGB] >> (12. A metric space (X,d) is said to be complete if every Cauchy sequence in X converges (to a point in X).