REAL ANALYSIS I

... it is useful to state t hem as a starting point for the study of real analysis and also to focus on one property, completeness, that is probably new to you. Field Properties The real number system ... gained from an introductory real analysis course. The book is designed to fil l the gaps left in the development of calculus as it is usually presented in an elementary course,...

Ngày tải lên: 08/03/2014, 12:20

... family F of intervals is said to cover the set S in R 1 if each point of S is in at least one of the intervals of F. A family F 1 is a subfamily of F if each member of F 1 is a member of F. For ... introduction to functions of bounded variation, we define the R-S integral and show how the usual integration-by-parts formula is a special case of this inte- gral. The generality...

Ngày tải lên: 31/03/2014, 15:12

... with functions, derivatives, and integrals. Here is a basic example of this. Let f be a real- valued function on R n which is continuously differentiable and has compact support, and fix a point x ∈ R n . ... years, in particular by Nabutovsky and Weinberger. See [NabW1, NabW2], for instance, and the references therein. 4 Quantitative topology, and calculus on sin- gular spaces...

Ngày tải lên: 31/03/2014, 16:28

... for instructors wishing to enrich their teach- ing with some carefully chosen problems and for individuals who are interested in solving difficult problems in mathematical analysis on the real axis. ... by topic into eight chapters, each of them containing both sections of proposed problems with complete solutions and separate sections including auxiliary problems, their so...

Ngày tải lên: 25/05/2014, 21:51

... Analysis III Real Analysis: Measure Theory, Integration, and Hilbert Spaces IV Functional Analysis: Introduction to Further Topics in Analysis 22 Chapter 1. MEASURE THEORY n, and define compact ... series and integrals. II. Complex analysis. III. Measure theory, Lebesgue integration, and Hilbert spaces. IV. A selection of further topics, including func...

Ngày tải lên: 12/06/2014, 16:36

... pretty ric h algebraic structure. In particular, it is a partially ordered linear space (see Chapter F). Wh en the dom ain of a real function is a poset, we can talk about how this map a ects the ... exists a function f : A → V A such that f (A) ∈ A for each A ∈ A. One can reword this in a few other wa y s. 12 But, how about the following algorithm? Start with A 1...

Ngày tải lên: 04/07/2014, 10:20

... countable for any m ∈ N. (b) Prov e or dispro ve: N ∞ is countable. Exercise 6.Let A and B be any sets such that A is countable, B is uncountable and A ⊆ B. Can B\ A = {x ∈ B : x/∈ A} be countable? ... irrational numbers countable? Exercise 7. H Let A and B be any sets such that A is countable and B is uncountable. Show that A ∪ B is uncountable. In fact, show that there is a b...

Ngày tải lên: 04/07/2014, 10:20

... finite? Exercise 28. H (a) Show that a separable metric space need not be compact, but a compact metric space X is separable. (b) Sho w that a connected metric space need not be compact, and a compact ... a metric space as compact. This is mostly due to the fact that some textbooks which focus exclu- sively on Euclidean spa c es define compactness as the combination o f the properties of clos...

Ngày tải lên: 04/07/2014, 10:20

... homeomorphism. (b)(Carothers) (X, d) and (X, D) may be homeomorphic, even if d and D are not equivalent. For instance, let X := {0, 1, 1 2 , 1 3 , } ,d: = d 1 , and de fine D on X 2 as follows: D( x, 1) := x and D( x, 0) := ... functions definedonanarbitrarymetric space. Let T be an y metric space, and recall that B(T ) stands for the set of all bounded functions defined on T, whichi...

Ngày tải lên: 04/07/2014, 10:20

... indeed a metric space. We also provide several o ther exercises here to help you get better acquain ted with the Haus- dor ff metric. (Fo r a more detailed analysis, see the first chapter of Nadler ... multifunctions, some many-valued maps, and still others refer to them as set-valued maps. In the economics literature, however, the term correspondence se ems widely agreed upon. 3 It is not...

Ngày tải lên: 04/07/2014, 10:20

... thin k of the sum of two real function s f and g defined on a n on emp ty set T as the real function f + g ∈ R T with (f + g)(t): =f( t)+g(t). Similarly, the product of λ ∈ R and f ∈ R T is the function ... tent of our fina l axiom on : The Monotonicit y Axiom ∗ . For any f, g ∈ H Ω,X ,iff ω  ∗ g ω for all ω ∈ Ω,then we have f  g. Moreov er, if f ω  ∗ g ω for all ω ∈ Ω, an...

Ngày tải lên: 04/07/2014, 10:20

... 1. So if L(x) > α w as the case, then we would have (1 − λ)L(x)+λL(y) > α for λ > 0 small enough, which would yield L(z λ ) > α (for small λ > 0 ), contradicting α >L(T). Thus ... a conv ex set S that con ta ins the origin in its algebraic in te rior is well-defin ed as a real function since, given that 0 ∈ al-int X (S),wehave  λ > 0: 1 λ x ∈ S  = ∅ for all x ∈ X. ......

Ngày tải lên: 04/07/2014, 10:20

... L X )withL (h( ω)) (which is a number) — you get L ◦ h. It is easy to see that there is a map ϕ on the set of all L◦ hs s uch that ϕ(L ◦ h) ≥ ϕ(L ◦ g) iff f  g — this is exactly what (12) s ays. The ... f,g H ,X , then there exists a (,u) L ìR X such th at  is represented by the map h → S ω∈Ω μ(ω)E h ω (u). What next? We ha ve weak ened the axiomatic system of the Anscom be-Aumann...

Ngày tải lên: 04/07/2014, 10:20

... that i t lies in? This seem in gly paradoxic al situation is just anoth e r illustration of h ow our finite dimensional intu- ition can go astray in the realm of infinite dimensional spaces. Indeed, ... nontrivial if X = {0}, finite dimensio n al if dim(X) < ∞, and infinite dimensional if dim(X)=∞. Put succinctly, a metric linear space is a linear space endowed with a translation invarian...

Ngày tải lên: 04/07/2014, 10:20

... and continuity in conjunction, along with a s atis fac tory geom etric analysis. When the norm under consideration is apparent from the context, it is customary to dispense with the notation (X, ... ? (Yes!) 480 Chapter J Norm ed Linear Spaces This chapter introduces a v ery im portan t subclass of metric linear spaces, namely, the class of normed linear space s. We begin with...

Ngày tải lên: 04/07/2014, 10:20