Often, if the metric dis clear from context, we will simply denote the metric space x. Spaces is a modern introduction to real analysis at the advanced undergraduate level. This metric is often called the euclidean or usual metric, because it is the metric that is suggested by euclidean geometry, and it is the most common metric used on r n. Upper and lower limits of sequences of real numbers, continuous functions, differentiation, riemannstieltjes integration, unifom convergence and applications, topological results and epilogue. The dual space e consists of all continuous linear functions from the banach space to the real numbers. A metric space is a set x together with a function d called a metric or distance function which assigns a real number dx, y to every pair x, y x satisfying the properties or axioms. Recall that a banach space is a normed vector space that is complete in the metric associated with the norm. N of real numbers is called bounded if there is a number. I introduce the idea of a metric and a metric space framed within the context of rn.
Let be a cauchy sequence in the sequence of real numbers is a cauchy sequence check it. If youve had a good real analysis course, then a lot though not all of the proofs below should. Field properties the real number system which we will often call simply the reals is. A metric space is a set in which we can talk of the. Real analysis on metric spaces mark dean lecture notes for fall 2014 phd class brown university 1lecture1 the. The following table contains summaries for each lecture topic listed. Browse other questions tagged real analysis metric spaces or ask your own question. Free and bound variables 3 make this explicit in each formula. Jan 22, 2012 this is a basic introduction to the idea of a metric space.
Xthe number dx,y gives us the distance between them. In the following we shall need the concept of the dual space of a banach space e. A metric space is, essentially, a set of points together with a rule for saying how far apart two such points are. For the purposes of boundedness it does not matter. Metric spaces are also a kind of a bridge between real analysis and general topology. Metric spaces are generalizations of the real line, in which some of the theorems that hold for r remain valid. Introduction to real analysis fall 2014 lecture notes. Click download or read online button to get metric space book now. The most familiar is the real numbers with the usual absolute value. A metric space is called complete if every cauchy sequence converges to a limit. If a subset of a metric space is not closed, this subset can not be sequentially compact. Completeness of the space of bounded real valued functions on a set, equipped with the norm, and the completeness of the space of bounded continuous real valued.
This site is like a library, use search box in the widget to get ebook that you want. The abstract concepts of metric ces are often perceived as difficult. A metric space is a set x where we have a notion of distance. Since is a complete space, the sequence has a limit. Difference between open sets and open balls in metric space. Metric space more examples on metric space in hindi. When dealing with an arbitrary metric space there may not be some natural fixed point 0.
It is also sometimes called a distance function or simply a distance. 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. Some of the main results in real analysis are i cauchy sequences converge, ii for continuous functions flim n. To allows an appreciation of the many interconnections between areas of mathematics. Compactness of metric spaces compactness in metric spaces the closed intervals a,b of the real line, and more generally the closed bounded subsets of rn, have some remarkable properties, which i believe you have studied in your course in real analysis. Introductory analysis i fall 2014 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. Some of this material is contained in optional sections of the book, but i will assume none of that and start from scratch. The particular distance function must satisfy the following conditions. Most of the spaces that arise in analysis are vector, or linear.
The fact that every pair is spread out is why this metric is called discrete. X of a metric space x,d in terms of properties of the corresponding metric subspace a,da. The metric structure in a normed space is very special and possesses many properties that general metric spaces do not necessarily have. Real analysis on metric spaces columbia university. As mentioned in the introduction, the main idea in analysis is to take limits. Real variables with basic metric space topology download book. Metrics on spaces of functions these metrics are important for many of the applications in. Find materials for this course in the pages linked along the left. Nov 08, 2014 a metric space is a set x together with a function d called a metric or distance function which assigns a real number dx, y to every pair x, y x satisfying the properties or axioms. This, instead of 8xx2rx2 0 one would write just 8xx2 0. Often d is omitted and one just writes x for a metric space if it is clear from the context what metric is being used. Feb 29, 2020 a subset of the real numbers is bounded whenever all its elements are at most some fixed distance from 0. However, not just any function may be considered a metric.
We briefly glance over the various kinds of metrics without too much attention to proofs. Then this does define a metric, in which no distinct pair of points are close. A space x is separable if it admits a countable dense subset. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc.
These notes are collected, composed and corrected by atiq ur rehman, phd. Metric spaces these notes accompany the fall 2011 introduction to real analysis course 1. Sometimes restrictions are indicated by use of special letters for the variables. Metric spaces are an abstraction generalizing the real line. A subset is called net if a metric space is called totally bounded if finite net. In chapter 2 we learned to take limits of sequences of real numbers.
Chapter 1 metric spaces these notes accompany the fall 2011 introduction to real analysis course 1. The rational numbers q with the usual metric is a complete metric space. To learn about the countability of sets, metric space, continuity, discontinuities, connectedness and compactness for set. Definition 1 a metric space m,d is a set m and metric d.
With every metric space there is associated a topology that precisely captures the notion of continuity for the given metric. Metric spaces notes these are updated version of previous notes. Jul 17, 2018 the function d is called the metric on x. Real analysismetric spaces wikibooks, open books for an. This course is concerned with the notion of distance.
It is forwardlooking in the sense that it first and foremost aims to provide students with the concepts and techniques they need in order to follow more advanced courses in mathematical analysis and neighboring fields. A metric space can be thought of as a very basic space having a geometry, with only a few axioms. Metric space download ebook pdf, epub, tuebl, mobi. A metric space y is clocally linearly connected if there exits c. 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. Metric spaces could also have a much more complex set as its set of points as well. A metric space is a set xtogether with a metric don it, and we will use the notation x. For example r is separable q is countable, and it is dense since every real number is a limit of rationals. The metric topology on a metric space m is the coarsest topology on m relative to which the metric d is a continuous map from the product of m with itself to the nonnegative real numbers. Introduction when we consider properties of a reasonable function, probably the.
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