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Maria Predoi Trandafir Blan

Craiova, 2005

VII

PREFACE

VIII R

Craiova, September 2005

IX

CONTENTS

VOL. I. DIFFERENTIAL CALCULUS

PREFACE VII

Chapter I. PRELIMINARIES

§I.1Sets. Relations. Functions1

Problems § I.1.11

§I.2Numbers15

Problems § I.2.26

§I.3Elements of linear algebra29

Problems § I.3.43

§I.4Elements of Topology47

Part 1. General topological structures47

Part 2. Scalar products, norms and metrics52

Problems § I.4.58

Chapter II. CONVERGENCE

§II.1Nets61

Part 1. General properties of nets61

Part 2. Sequences in metric spaces65

Problems § II.1.74

§II.2Series of real and complex numbers77

Problems § II.2.89

§II.3Sequences and series of functions93

Problems § II.3.111

§II.4Power series117

Problems § II.4.130

X

Chapter III. CONTINUITY

§III.1Limits and continuity in135

Problems § III.1.142

§III.2Limits and continuity in topological spaces144

Problems § III.2.155

§III.3Limits and continuity in metric spaces156

Problems § III.3.163

§III.4Continuous linear operators165

Problems § III.4.179

Chapter IV. DIFFERENTIABILITY

§IV.1Real functions of a real variable183

Problems § IV.1.187

§IV.2Functions between normed spaces188

Problems§ IV.2.200

§IV.3Functions of several real variables201

Problems § IV.3.219

§IV.4Implicit functions224

Problems § IV.4.241

§IV.5Complex functions245

Problems § IV.5.260

INDEX264

BIBLIOGRAPHY270

1

CHAPTER I. PRELIMINARIES

§I.1. SETS, RELATIONS, FUNCTIONS

From the very beginning, we mention that a general knowledge of set theory is assumed. In order to avoid the contradictions, which can occur in such a "naive" theory, these sets will be consideredof a total set, i.e. elements of(). The sets are usually depicted by some specific properties of the component elements, but we shall take care that instead of it is advisable to speak of (see [RM], [SO], etc). When operate with sets we basically need one unary operation two binary operations and a binary relation (i)‰‰‰‰ ˆˆˆˆ() (ii)‰ˆ‰ˆ‰ ˆ‰ˆ‰ˆ (iii)ˆ‰ ‰ˆ() (iv)ˆ{‰ ‰{ˆ() (v)‰‰ ˆˆ()

1.2.Remark.From the above properties (i)-(v) we can derive the whole set

theory. In particular, the associativity is useful to define intersections and unions of a finite number ofsets, while the extension of these operations to are frequent, e.g.‡=ˆ{for the (unique!) ˆ{for the ' ‰for the Ž (defined by‰ )for the relation ofetc.

Chapter I. Preliminaries

More generally, a non-void seton which the equality = , and the operationsº,'andš(instead of{,‰respectivelyˆ) are defined, such that conditions (i)-(v)hold as axioms, represents a Besides(), we mention the following important examples of Boolean algebras: the algebra of propositions in the formal logic, the algebra of switch nets, the algebra of logical circuits, and the field of events in a random process. The obvious analogy between these algebras is based on the correspondence of the following facts: -a set may contain some given point or not; -a proposition may be true or false; -an event may happen in an experience or not; -a switch may let the current flow through orbreak it; -at any point of a logical circuit may be a signal or not. In addition, the specific operations of a Boolean algebra allow the following concrete representations in switch networks: Similarly, in logical circuits we speak of"logical gates" like A A Ae eA(double switch) A

BA_B(parallel connection)

ABAB^(serial connection)

AAee( -gate)

A ABBC C ___( -gate) A ABB C

C^^^( -gate)

§ I.1. Sets, Relations, Functions

1.3.The Fundamental Problemsconcerning a practical realization of a

switch network, logic circuits, etc., are theand the. In the first case, we have some physical realization and we want to know how it works, while in the second case, we desire a specific functioning and we are looking for a concrete device that should work like this. Both problems involve the so-called , which describe the functioning of the circuits in terms of values of agiven formula, as in the table from bellow. It is advisable to start by putting the values ...forthen foretc., under these variables, then continue by the resulting values under the involved connectorsš,',o, etc. by respecting the order of operations, which is specified by brackets. The last completed column, which also gives the name of the formula, contains the "truth values" of the considered formula. As for example, let us consider the following disjunction, whose truth- values are in column: (š)'[(ºo) +] where etc. show the order of completing the columns. The converse problem, namely that of writing a formula with previously given values, makes use of someexpressions, which equalonly once (called ). For example, if a circuit should function according to the table from below,

Chapter I. Preliminaries

then one working function is

This form ofis called (see [ME], etc.).

The following type of subfamilies of(), where‡z, is frequently met in the Mathematical Analysis (see [BN1], [DJ], [CI], [L-P], etc.):

1.4.Definition.A nonvoid family() is called ()if

[F0]‡; Sometimes condition [F0] is omitted, and we speak of filters in generalized () sense. In this case,= If familyis a filter, then any subfamilyŽfor which (in particularitself) is called . = {ŽR:H> 0 such that'(-H, H)}, thenis a filter, and a base ofis= {(-H, H):H>0}. It is easy b) The family(N), defined by c) Let(R2) be thefamily of interior parts of arbitrary regular is a filterfor which family, of all interior parts of the disks centered at ( ), is a base (as well asitself).

1.6.Proposition.In an arbitrary total setz‡we have:

Any baseof a filterŽ() satisfies the condition

IfŽ() satisfies condition [FB] (i.e. together with [F0] it is a ), then the family of oversets is a filter in(); we say that filteris .

Ifis a base of, thengenerates.

§ I.1. Sets, Relations, Functions

The proof is direct, and we recommend it as an exercise.

1.7.Definition.Ifandare nonvoid sets, their is

Any partRŽuis called In

particular, ifRŽuit is named For example,the

IfRis a relation on, itsis defined by

of two relationsRandSonis noted ()ofRatis defined by

Most frequently, a binary relationRonmay be:

:GŽR; :R=R-1; :RˆR-1=G; :RRŽR; The reflexive, symmetric and transitive relations are called, and usually they are denoted bya. Ifais an equivalence on, then each The set of all equivalence classes is called and it is noted/a.

The reflexive and transitive relations are named.

Any antisymmetric preorder is said to be a , and usually it is denoted byd. We say that an orderdonis(or, equivalently, (,d) dFinally,(,d) is said to be (ordis a on ) iffdis total and any nonvoid part ofhas a smallest element.

1.8.Examples.Equivalences:

1.The equality (of sets, numbers, figures, etc.);

Chapter I. Preliminaries

The similarityof the figures (triangles, rectangles, etc.) inR2,R3, etc., is an equivalence especially studied in Geometry.

Orders and preorders:

1.The inclusion in() is a partial order;

2.N,Z,QandRare totally ordered by their natural ordersd;

3.Nis well ordered by its natural order;

4.Ifis (totally) ordered byd, andis an arbitrary nonvoid set, then the

setFof all functions:o, is partially ordered by This relation is frequently called (compare to the examples in problem 9, at the end of the paragraph). Directed sets(i. e. preordered sets (,d) with directedd):

1.(N,d) , as well as any totally ordered set;

2.Any filter(e.g. the entire(), each system of neighborhoods()

in topological spaces, etc.) is directed by inclusion, in the sense that diffŽ. The pair (,d) is a directed set if the preorderdis defined by (,)d(,)oeŽ The same construction is possible using neighborhoods ...of a fixed point0in any topological space.

4.The,which occur in the definition of some integrals, generate

directed sets (see the integral calculus). In particular, in order for us to define the Riemannian integral on [ ]R, we considerof the closed interval [ ], i.e. finite sets of subintervals of the form

G= {],[1:= 1, 2, ..., n;=<<...< },

It is easy to see that setof all pairs (GG)), is directed by relation d, where (GïGïd(GïïGïïLIIGïŽGïï There is a specific terminology in preordered sets, as follows:

1.9.Definition.Letbe a part ofwhich is (partially) ordered byd. Any

§ I.1. Sets, Relations, Functions

there exists one, then it is unique!), and we note0=max. If the set of all upper bounds ofhas the smallest element, we say that is the , and we note=sup. greater than*(the elementmaxif it exists, is maximal, but the converse assertion is not generally true). Similarly, we speak of , smallest element (denoted asmin), infimum (notedinf), and . Ifsupandinfdo exist for each bounded set, we say that (,d) is a(in order). Alternatively, instead of using an orderR, we can refer to the attached sometimes named (especially because ofits shape). If a partofis totally (linearly) ordered by the induced order, then we say thatis ain. there existinf{ } =šandsup{ } ='If the infimum and the supremum exist for any bounded set in, then the lattice is said to be (or1). A remarkable example of lattice is the following:

1.10.Proposition.Every Boolean algebra is a lattice. In particular,() is

a (complete) lattice relative toŽ Proof.We have to show thatŽis a (partial) order on(), and each for an arbitrary Boolean algebra, reflexivity ofŽmeans A'A=A. In fact, according toandin proposition 1.1., we have

A'A = [š(')]'[š(')] = (')š(') ='(š) =

Fromdanddwe deduce that ' hencedis

antisymmetric. For transitivity, ifdanddwe obtaindsince according to d'andd'On the other hand, ifdand d, we haveš andš so that i.e.'dSimilarly we can reason forinf{} =šas well as for arbitrary families of sets in.}

1.11.Remark.The above proof is based on the properties, hence it

is valid in arbitrary Boolean algebras. If limited to(), we could reduce it to the concrete expressions of‰ ˆ Žetc. According to the Stone's theorem, which establishes that any Boolean algebrais

Chapter I. Preliminaries

isomorphic to a family of parts, verifying a property inas for() is still useful.

1.12.Definition.Letandbe nonvoid sets, andRŽube a relation

between the elements ofand. We say thatRis adefined on with values iniff the sectionR[x] reduces to a single element offor :etc.

We say that:is(1:1, i.e.) iffz

wheneverz (or). Ifis both injectiveand surjective, it is called (1:1 map ofon, or 1:1 ) Any function:can be extended to() and() by considering

Ž, defined by

and Ž, defined by Ifis bijective, thenmconsists of a single element, so we can speak of defined by oe

If:oand:othen:odefined by

is called and we note

The :ois a part ofu namely

On a Cartesian productuwe distinguish two remarkable functions, called, namelyPr:uo, andPr:uo, defined by

Pr(,) =, andPr(,) =.

In the general case of an arbitrary Cartesian product, which is defined by we get aPr:(

Pr:() =() .

Sometimes we must extend the above notion of function, and allow that consists of more points; in such case we say thatis a(or ). For example, in the complex analysis, is supposed to be an already knownfunction. Similarly, we speak of , or functions.

§ I.1. Sets, Relations, Functions

This process of extending the action ofcan be continued to carry elements from(()) to(()), e.g. ifŽ(), then

1.13.Examples.Each partŽ(z‡)is completely determined by its

:o{0, 1}expressed by A.x1 Axif0 In other terms,() can be presented as the set of all functions defined on and taking two values. Because we generally note the set of all functions :oby, we obtain(). We mention that this possibility to represent sets by functions has led to the idea of , having characteristic functions with values in the closed interval [0,1] ofR(see [N-R], [KP], etc.). Formally, this means to replace() by [0,1]. Of course, when we work with fuzzy sets, we have to reformulate the relations and the operations with setsin terms of functions, e.g.Žas fuzzy sets meansdas functions,{= 1-, },{max ‰,},{min ˆ, etc.

1.14.Proposition.Let:obe a function, and let be arbitrary

m({){[m()]holds for anyŽ, while({)and{[()] generally cannot be compared.

The proof is left to the reader.

The following particular type of functions is frequently used in the

Mathematical Analysis:

1.15.Definition.Letbe a nonvoid set. Any function:Nois called

the sequenceby mentioning the generic term()

A sequence:Nois considered to be a iff

for some increasing:NoN(i.e.dŸd). Usually we note so that a subsequence of()takes the form)( More generally, if (d) is a directed set, then:ois called (briefly, or) inInstead of, the g.s. is

Chapter I. Preliminaries

orofiff whereofulfils the following condition (due to Kelley, see [KJ],[DE], etc.):quotesdbs_dbs21.pdfusesText_27

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