The particular distance function must satisfy the following conditions. We do not develop their theory in detail, and we leave the veri. Informally, 3 and 4 say, respectively, that cis closed under. Since every continuous function on a closed and bounded interval is bounded, therefore we have i i i i. Rather than passing quickly from the definition of a metric to the. A metric space is a set xtogether with a metric don it, and we will use the notation x. Spaces is a modern introduction to real analysis at the advanced undergraduate level. Rather than passing quickly from the definition of a metric to the more abstract concepts of convergence and continuity, the author takes the concrete notion of distance as far as possible. The aim is to give a very streamlined development of a course in metric space topology emphasizing only the most useful concepts, concrete spaces and geometric ideas. The fact that every pair is spread out is why this metric is called discrete. Variety of examples along with real life applications have been provided to understand and appreciate the beauty of metric spaces. A metric space is a pair x, d, where x is a set and d is a metric on x.
Real analysis on metric spaces mark dean lecture notes for fall 2014 phd class brown university 1lecture1 the. In some contexts it is convenient to deal instead with complex functions. This, instead of 8xx2rx2 0 one would write just 8xx2 0. Notes on metric spaces these notes introduce the concept of a metric space, which will be an essential notion throughout. Any normed vector space can be made into a metric space in a natural way. These notes are collected, composed and corrected by atiq ur rehman, phd. This volume provides a complete introduction to metric space theory for undergraduates. 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. If x,d is a metric space and a is a nonempty subset of x, we can make a metric d a on a by putting. To encourage the geometric thinking, i have chosen large number of examples which allow us to draw pictures and develop our intuition and draw conclusions, generate ideas for proofs. It covers the topology of metric spaces, continuity, connectedness, compactness and product spaces, and includes results such as the tietzeurysohn extension theorem, picards theorem on ordinary differential equations, and the set of discontinuities of the pointwise limit of a sequence of continuous functions. Real variables with basic metric space topology this is a text in elementary real analysis. Several concepts are introduced, first in metric spaces and then repeated for topological spaces, to. Metric space download ebook pdf, epub, tuebl, mobi.
Lecture notes assignments download course materials. Every metric space is a topological space in a natural manner, and therefore all definitions and theorems about general topological spaces also apply to all metric spaces. The abstract concepts of metric spaces are often perceived as difficult. Real analysismetric spaces wikibooks, open books for an. This book is about metric spaces of nonpositive curvature in the sense of busemann, that is, metric spaces whose distance function satisfies a convexity condition. If v,k k is a normed vector space, then the condition du,v ku. A very basic example of a metric space is the real line rwith its standard metric. The most familiar is the real numbers with the usual absolute value. Nearly all the concepts we discuss for metric spaces are natural generalizations of the corresponding concepts for r with this absolutevalue metric. About any point x \displaystyle x in a metric space m \displaystyle m we define the open ball of radius r 0 \displaystyle r0 where r \displaystyle r is a real. Since is a complete space, the sequence has a limit. Upper and lower limits of sequences of real numbers, continuous functions, differentiation, riemannstieltjes integration, unifom convergence and applications, topological results and epilogue.
Often, if the metric dis clear from context, we will simply denote the metric space x. Introduction to real analysis fall 2014 lecture notes. Metric spaces joseph muscat2003 last revised may 2009 a revised and expanded version of these notes are now published by springer. 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. For all of the lecture notes, including a table of contents, download the following file pdf. These are actually based on the lectures delivered by prof. For all of the lecture notes, including a table of contents, download the following file pdf 1. Metric spaces could also have a much more complex set as its set of points as well. Rather than passing quickly from the definition of a metric to the more abstract concepts of convergence and continuity, the author takes the concrete notion of.
Some of this material is contained in optional sections of the book, but i will assume none of that and start from scratch. Real variables with basic metric space topology dover. However, not just any function may be considered a metric. Introduction to metric and topological spaces oxford. U nofthem, the cartesian product of u with itself n times. Proofs, sets, and functions, the foundation of calculus, metric spaces, spaces of continuous functions, modes of continuity, applications to differential equations, applications to power series.
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. Lecture notes analysis ii mathematics mit opencourseware. Mathematical proof or they may be 2place predicate symbols. Real variables with basic metric space topology download. Nov 22, 2012 we discuss the introduced concept of g metric spaces and the fixed point existing results of contractive mappings defined on such spaces. Let be a cauchy sequence in the sequence of real numbers is a cauchy sequence check it.
Metric space topology, as the generalization to abstract spaces of the theory of sets of points on a line or in a plane, unifies many branches of classical analysis and is necessary introduction to functional analysis. In calculus on r, a fundamental role is played by those subsets of r which are intervals. Sometimes restrictions are indicated by use of special letters for the variables. This book offers a unique approach to the subject which gives readers the advantage of a new perspective familiar from the analysis of a real line. Chapter 1 metric spaces islamic university of gaza.
Professor copsons book, which is based on lectures given to thirdyear undergraduates at the university of st andrews, provides a more leisurely treatment of metric spaces than. Topics include metric spaces, euclidean spaces and their basic topological properties, sequences and series of real numbers, continuous functions, differentiation, riemannstieltjes integration, and uniform convergence and applications. It is also sometimes called a distance function or simply a distance. A brief guide to metrics, norms, and inner products. A metric space is called complete if every cauchy sequence converges to a limit. Remarks on g metric spaces and fixed point theorems. In particular, we show that the most obtained fixed point theorems on such spaces can be deduced immediately from fixed point theorems on metric or quasimetric spaces. In particular, we show that the most obtained fixed point theorems on such spaces can be deduced immediately from fixed point theorems on metric or quasi metric spaces. The language of metric and topological spaces is established with continuity as the motivating concept. A good book for metric spaces specifically would be o searcoids metric spaces. However, note that while metric spaces play an important role in real analysis, the study of metric spaces is by no means the same thing as real analysis.
Upper and lower limits of sequences of real numbers, continuous functions, differentiation, riemannstieltjes integration, unifom convergence. These are the notes prepared for the course mth 304 to be o ered to undergraduate students at iit kanpur. If a page of the book isnt showing here, please add text bookcat to the end of the page concerned. Chapter 9 the topology of metric spaces uci mathematics. Topics include metric spaces, open and closed sets, convergent sequences, function limits and continuity, compact sets, sequences and series of functions, power series, differentiation and integration, taylors theorem, total variation, rectifiable arcs, and sufficient conditions of integrability. 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.
Real variables with basic metric space topology download book. We discuss the introduced concept of gmetric spaces and the fixed point existing results of contractive mappings defined on such spaces. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc. The book also contains a systematic introduction to the theory of geodesics in metric spaces, as well as a detailed presentation of some facets of convexity theory that are useful in the study of nonpositive curvature. Denote then since is a cauchy sequence, rewrite it as this means that and metric spaces metric spaces page 1. Once the terms have been speci ed, then the atomic formulas are speci ed. Metrics on spaces of functions these metrics are important for many of the applications in. It is not hard to check that d is a metric on x, usually referred to as the discrete metric. There are many ways to make new metric spaces from old. Then this does define a metric, in which no distinct pair of points are close.
Review of metric spaces and pointset topology september 29, 2016 an open set in r n is any set with the property observed in the latter corollary, namely a set uin r n is open if for every xin uthere is an open ball centered at xcontained in u. Real variables with basic metric space topology dover books. This awardwinning text carefully leads the student through the basic topics of real analysis. It is also sometimes called a distance function or simply a distance 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 we already know a few examples of metric spaces. The analogues of open intervals in general metric spaces are the following. Remarks on g metric spaces and fixed point theorems fixed. The lecture notes were taken by a student in the class. Chapter 1 metric spaces these notes accompany the fall 2011 introduction to real analysis course 1. Xthe number dx,y gives us the distance between them. Then we call k k a norm and say that v,k k is a normed vector space. Spaces an introduction to real analysis download book. This book offers a unique approach to the subject which gives readers the advantage of a new perspective on ideas familiar from the analysis of a real line. Detailed solutions to the problems appear within the text, making this volume ideal for independent study.
Metric spaces are generalizations of the real line, in which some of the theorems that hold for r. Moreover the concepts of metric subspace, metric superspace. You can view a list of all subpages under the book main page not including the book main page itself, regardless of whether theyre categorized, here. The aim is to move gradually from familiar real analysis to abstract topological spaces, using metric spaces as a bridge between the two. A good book for real analysis would be kolmogorov and fomins introductory real analysis. Then d is a metric on r2, called the euclidean, or. Ais a family of sets in cindexed by some index set a,then a o c. Free and bound variables 3 make this explicit in each formula. A metric space is a set x where we have a 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. Metricandtopologicalspaces university of cambridge. Metric spaces notes these are updated version of previous notes.
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