Norms and metrics, normed vector spaces and metric spaces. Lecture notes analysis ii mathematics mit opencourseware. Mar 05, 2018 in this video, we have discussed the metric space with examples on metric space namaste to all friends, this video lecture series presented by vedam institute of mathematics is useful to all. Chapter 2 metric spaces the purpose of this chapter is to present a summary of some basic properties ofmetric and topological spaces that play an important role in the mainbody of the book. Introduction to real analysis fall 2014 lecture notes.
You have met or you will meet the concept of a normed vector space both in algebra and analysis courses. Lecture notes, lecture 17 brief notes metric spaces. A metric space consists of a set xtogether with a function d. Chapter 3 starts with some standard facts about metric spaces and relates the concepts to measure theory. Well generalize from euclidean spaces to more general spaces, such as spaces of functions. Introduction to real analysis spring 2014 lecture notes. A metric space is a pair x,d where x is a nonempty set and d is a function d. Notes on metric spaces these notes are an alternative to the textbook, from and including closed sets and open sets page 58 to and excluding cantor sets page 95 1 the topology of metric spaces assume m is a metric space with distance function d. Nearly all the concepts we discuss for metric spaces are natural generalizations of the corresponding concepts for r with this absolutevalue metric. For example, the rational numbers are a metric space, and any sequence of rationals that converges to an irrational number in r is a cauchy sequence in q but has no limit in q. Notes of metric spaces these notes are related to section iv of b course of mathematics, paper b. Cauchy sequences do not converge in all metric spaces. Also any subset of a metric space is a metric space.
We usually denote sn by s n, called the nth term of s, and write fs ngfor the sequence, or fs 1. A sequence in a set xa sequence of elements of x is a function s. Though these notes are mainly on metric spaces, we have taken the attitude that if a notion is the same for metric spaces and the more general topological spaces, then we use the more general setting. The axiom m2 says that a metric is symmetric, and the axiom m3 is called. Similarly, if a proof of a result is more or less the same for metric and topological spaces, we give the proof in full generalty. Lecture notes on metric space and gromovhausdor distance. I can send some notes on the exercises in sections 16 and 17 to supervisors by email. Metric spaces a metric space is a set x that has a notion of the distance dx,y between every pair of points x,y. One such example is the 4point equilateral space, with every two points at distance 1. A metric space x,d is complete if and only if every nested sequence of nonempty closed subset of x. Lecture notes from the integration workshop at university of arizona, august 2018. Definition 1 a metric space m,d is a set m and metric d. Oxfordsmathematics lecture notes, particularly notes on m2 analysis, m1 groups, a2 metric spaces, a3 rings and modules, a5 topology, and aso groups. Metric space more examples on metric space in hindilecture.
When we discuss probability theory of random processes, the underlying sample. Uniform metric let be any set and let define particular cases. These are the notes prepared for the course mth 304 to be o ered to undergraduate students at iit kanpur. R the set of real valued continuous functions on the interval, with l p metric df. The concepts discussed form a foundation for an undergraduate programme in mathematical analysis.
Lecture notes assignments download course materials. Then d is a metric on r2, called the euclidean, or. This note tries to recap this excellent introduction course. A metric space is a set x where we have a notion of distance. Lecture 3 complete metric spaces 1 complete metric spaces 1. A metric space is called complete if every cauchy sequence converges to a limit. These notes are collected, composed and corrected by atiq ur rehman, phd. Additional lecture notes for math 63cm, version 2020. This page relates to math3901 metric spaces as it was given in the year 2000.
A where a is an open subset of rn, and fa, x and, 1. Some of this material is contained in optional sections of the book, but i will assume none of that and start from scratch. This generalization of the absolute value on ror c to the framework of vector spaces is central to modern analysis. These notes are based heavily on notes from previous integration workshops written by philip foth, tom kennedy, shankar venkataramani and others. Please note, the full solutions are only available to lecturers.
In fact, the definition for functions on r can be easily adjusted so that it applies to functions on an arbitrary metric space. Real analysis on metric spaces columbia university. This notes is based on the lecture given by ilaria mondello for the 20172018 master day of parissaclay. Note that iff if then so thus on the other hand, let. The particular distance function must satisfy the following conditions. Summer 2007 john douglas moore our goal of these notes is to explain a few facts regarding metric spaces not.
This compilation has been made in accordance with the. Complete metric spaces a metric space x is complete if. Moreover the concepts of metric subspace, metric superspace, isometry i. I have put together for you in this book an introductionalbeit a fairly thorough introductionto metrics and metric spaces.
Metric spaces notes these are updated version of previous notes. Normed vector spaces and metric spaces were going to develop generalizations of the ideas of length or magnitude and distance. Introduction let x be an arbitrary set, which could consist of vectors in rn. The aim of these lectures notes is to provide a gentle introduction to the theory of gradient ows in metric spaces developed in the rst part of the book of ambrosiogiglisavar e ags. In this video, we have discussed the metric space with examples on metric space namaste to all friends, this video lecture series presented by vedam institute of mathematics is. Xthe number dx,y gives us the distance between them. Namely, we will discuss metric spaces, open sets, and closed sets. Lecture notes on metric and topological spaces niels. Lecture notes for fall 2014 phd class brown university.
Chapter 1 metric spaces these notes accompany the fall 2011 introduction to real analysis course 1. Wesaythatasequencex n n2n xisacauchy sequence ifforall0 thereexistsann. A metric space is, essentially, a set of points together with a rule for saying how far apart two such points are. For all of the lecture notes, including a table of contents, download the following file pdf 1. Metricandtopologicalspaces university of cambridge. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc. Metric space more examples on metric space in hindi. Notes on metric spaces these notes introduce the concept of a metric space, which will be an essential notion throughout this course and in others that follow. Two metric spaces are isometric if there exists a bijective isometry between them.
Additional lecture notes for math 63cm, version 2020 lenya ryzhik january 14, 2020 1 complete metric spaces 1. A pair, where is a metric on is called a metric space. However, metric spaces are somewhat special among all shapes that appear in mathematics, and there are cases where one can usefully make sense of a notion of closeness, even if there does not exist a metric function that expresses this notion. Can choose a metric suited to particular purpose metrics may be complicated, while the topology may be simple can study families of metrics on a xed topological space. To register for access, please click the link below and then select create account. Summer 2007 john douglas moore our goal of these notes is to explain a few facts regarding metric spaces not included in the. Metric space more examples on metric space in hindi lecture 2 duration. We do not develop their theory in detail, and we leave the veri. Imus lecture notes on harmonic analysis, metric spaces and pdes, sevilla 2011 edited by.
Lecture notes on topology for mat35004500 following j. We begin with the familiar notions of magnitude and distance on the real line. Sutherland partial results of the exercises from the book. A metric space is just a set x equipped with a function d of two variables.
For all of the lecture notes, including a table of contents, download the following file pdf. U nofthem, the cartesian product of u with itself n times. Partial solutions are available in the resources section. Lucas cha ee, taylor dupuy, rafael espinosa, jarod hart, anna kairema, lyudmila korobenko. Imus lecture notes on harmonic analysis, metric spaces and. Note that this definition exactly mimics the definition of convergence in. The lecture notes were taken by a student in the class. The spectral theorem 105 these are lecture notes that have evolved over time. Note that so it is closed as a compliment of an open set. Metric spaces lecture notes semester 1, 2007 notes by.
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