Appendix A Construction of Real Numbers

The purpose of this appendix is to give a construction of the field lR of real numbers from the field iQ of rational numbers which, we assume, is known to the reader. Let us point out that we did not give the axiomatic construction of the set N of natural numbers from which one can first construct the set Z of integers and, subsequently, the set iQ of rationals. These constructions may be found in most textbooks on abstract algebra, e.g., A Survey of Modern Algebra, by Birkhoff and MacLane [BM77]. Most authors use the so-called Dedekind Cuts to construct the set of real numbers from that of rational numbers. Since, however, the reader is now familiar with sequences and series, it is more natural to use Georg Cantor's method of construction, which is based on Cauchy sequences of rational numbers, and can be extended to more abstract situations. This abstraction, which is referred to as the completion of a metric space, was discussed in Chapter 5. We begin our discussion by introducing some notation and definitions.

Notation. We recall that the set of rational numbers is denoted by iQ and the set of positive rationals by iQ+ = {r E iQ : r > O}. Also, the set of all sequences of rational numbers, i.e., the set of all functions from N to iQ, is denoted by iQN . Next, we define the Cauchy sequences of rational numbers. Although the def­ inition of Cauchy sequences was given earlier, since we have not yet constructed the set of real numbers, we must insist that the e > 0 in our definition take rational values only. Definition A.I (Cauchy Sequences in iQ). A sequence x E iQN is called a Cauchy sequence if the following holds: 532 Appendix A. Construction of Real Numbers

The set of all Cauchy sequences in Q will be denoted by C. Next, we define null sequences in Q. Again, this was defined earlier, but we must be careful to use only rational e > 0 . Definition A.2 (Null Sequences in Q). A sequence x E ~ is called a null sequence if the following holds :

The set of all null sequences in Q will be denoted by N. Remark. Note that N is precisely the set of all rational sequences that converge to zero, and that we obviously have N c C. (Why?) As we have seen, the set Q of rationals is dense in the set lR of real numbers, which we introduced axiomatically. It follows that each real number { is the limit of a (not unique) sequence (x n ) of rational numbers. It is tempting, therefore, to take such a sequence (x n ) as the definition of the real number {. The non­ uniqueness of (x n ) poses a problem, however, for two such sequences in fact represent the same {. This motivates the following definition. Definition A.3 (Equivalent Cauchy Sequences). We say that two Cauchy sequences x , y E C are equivalent, and write x 'V y, if and only if x - yEN.

Exercise A.!. Show that the relation 'V is indeed an equivalence relation on the set C.

Notation. For each sequence x E C, its equivalence class is denoted by [x], and, we recall, is defined by [z] = {y E C: y 'V x}. The set of all equivalence classes of elements of C is denoted by C/N. Definition A .4 (Real Number). The set R of real numbers is defined to be lR := C/N. Thus { is a real number if {= [xl for some x E C. The sequence x E C is then called a representative of {. Clearly, if x and y both represent {, then x-yEN. Exercise A.2. (1) Show that a Cauchy sequence in Q is bounded. (2) Show that C is closed under addition and multiplication; i.e., 'rIx , y E C, we have x + y, xy E C. (3) Show that N is an ideal in C; i.e., it is closed under addition and satisfies the stronger condition that 'rIx E Nand 'rIy E C we have xy E N. Hints: For the addition, use an c/2-argument. For the multiplication, use the inequalities IXmYm - xnYnl ~ IYmllxm - xnl + IxnllYm - Ynl ~ Blxm - xnl + AIYm - Ynl, for some constants A, BE Q+ , where the second inequality follows from part (1). Definition A.5 (Addition, Subtraction, Multiplication). Let { = [xl and "I = [y] be any real numbers. We define { + "I, -"I, {- "I, and {"I (or { . "I) as follows: (1) {+ "I := [x + y], (2) -"I := [-y], (3) { - "I := { + (-"I) = [x - y], and (4) {"I := [xy]. Appendix A. Construction of Real Numbers 533

Exercise A.a. Show that the definitions of e + 11 and e11 are independent of the representatives x and y of e and 11, respectively. In other words, show that, if x rv x' and y rv s', then we have x + y rv x' + y' and xy rv x'y'. Hint: You will need arguments similar to those needed in Exercise A.2.

Proposition A.I (Ring Properties of R). The set R of real numbers is a commutative ring with identity. In other words, for all real numbers e, 11, and (, we have (1) e + 11 = 11 + e, (2) (e + 11) + ( = e + (11 + (), (9) 30 ER with 0 +e= e, (4) 3 - eE R with e+ (-e) = 0, (5) e11 = TJe, (6) (eTJ)( = e(TJ(), (7) 3 1 E R, 1 =1= 0, with 1 . e= e, and (8) e(TJ +() = eTJ + e(·

Proof. The proofs of these properties are straightforward. For example, to prove (2), note that if e= [x], TJ = [y], and ( = [z], then (e +TJ)+( = [(x+y) +z], while e+(TJ+() = [x+(y+z)]. Since we obviously have (x+y)+z = x+(y+z) in C, (2) follows. Note that the additive identity ("0" in (3)) is in fact 0 = [(0,0,0, .. . )] E N, and that the multiplicative identity ("I" in (7)) is 1 = [(1,1,1, )]. Also, 1 =1= 0 is obvious, because the sequences (0,0,0, . .. ) and (1,1,1, ) are not equivalent. 0

Proposition A.2. Let ¢ : Q -t R be defined by ¢(r) = [(r, r, r, .. . )]. Then ¢ is an injective "ring homomorphism". In other words, ¢ is a one-to-one map satisfying ¢(r + s) = ¢(r) + ¢(s), ¢(rs) = ¢(r)¢(s), ¢(O) = 0, and ¢(1) = 1, Vr, SEQ.

Exercise A.4. Prove Proposition A.2.

Remarks. By Proposition A.2, the map ¢ is a field isomorphism of Q onto its image ¢(Q) C Rj i.e., a one-to-one correspondence between Q and ¢(Q) that preserves all the algebraic properties of Q. Therefore, we henceforth identify the two sets and, by abuse of notation, will write Q = ¢(Q) CR. Based on this identification, the field Q of rational numbers becomes a subfield of the field R of real numbers. Here , by a field we mean a set F together with two operations "+" of addition and «.» of multiplication, i.e., two maps + : (x, y) t---+ X + y and· : (x, y) t---+ z . y, from F x F to F, satisfying the nine (algebraic) axioms (Ai - A4' M i - M 4 , D) stated for real numbers in section 1 of Chapter 2. Proposition A.l only shows that R is a commutative ring with identity. To prove that R is actually a field, the only property we need to check is the existence of reciprocals for nonzero real numbers [cf, Axiom (M4) at the beginning of Chapter 2). To this end, we shall need the following

Proposition A.a. Let ebe a nonzero element ofR. Then, there exists a rational number r E Q+ and a representative z E C ofesuch that, either X n ;::: r "In E N or Xn $ -r "In EN. 534 Appendix A. Construction of Real Numbers

Proof. Let Y EC be a representative of {. Since { i= 0, the sequence (Yn) is not equivalent to (O, 0, 0, ... ) and we have

(3c E Q+){'tIN E N){3n ~ N)(lYn - 01 ~ c).

On the other hand, (Yn) E C implies that

(3K E N)(m, n ~ K ~ IYm - Ynl < c/2).

Now, by (*), we can find k ~ K such that \Ykl ~ c. Changing {to -{, if necessary, we may assume that Yk ~ c. Therefore, using (**),

Let X n := c/2 for n < K, and X n = Yn for n ~ K. It is then clear that { = [(xn)), and that, with r := c/2, we have X n ~ r for all n E N. 0 Definition A.6 (Positive and Negative Cauchy Sequences). We say that a sequence x E C is positive (resp., negative) if it satisfies the first (resp., second) alternative in Proposition A.3. The set of all positive (resp., negative) sequences in C is denoted by C+ (resp. , C-). Remarks. It is obv ious that the two alternatives in Proposition A.3 are mutually exclusive; i.e., that C+ n C- = 0. Moreover, the cond ition in the first (and hence also second) alternative needs only be satisfied ultimately; i.e., it can be replaced by (3r E Q+)(3N E N)(n ~ N ~ X n ~ r) . Indeed, one can always replace x by the equivalent sequence x' defined by x~ := r 'tIk < N, and x~ := Xk 'tIk ~ N. Proposition A.4. We have C = C+ u N u C- , where the union is disjoint. In other words, {C+,N,C-} is a partition ojC. Proof. This is an obvious consequence of Proposition A.3. 0 We are now going to prove that lR is indeed a field.

Theorem A.I. The set lR of real numbers is a field. In other words, in addition to the ring properties {1}-{8} of Proposition A.l, we also have the following:

{'tI~ E lR\ {0})(3 1/~ E lR\ {O})(~ . (1/~) = 1). Proof. Suppose that ~ E lR \ {O}. By Proposition A.3, we can then find r E Q+ and a representative (xn ) of ~ such that Ixnl ~ r 'tin E N. If we can show that (l/xn ) E C, then, setting 1/{ := [(l/xn )), we clearly get ~ . (1/~) = 1. However , (x n ) EC implies

2 ('tic E Q+)(3N E N)(m, n ~ N ~ IXm - xnl < cr ).

Therefore,

2 IX m - xnl cr m, n ~ N ~ 11/xm - l/xnl = Ixmllxnl < -;:2 = e, which proves indeed that (l/xn ) EC and completes the proof. 0 Appendix A. Construction of Real Numbers 535

Having established the field properties of JR, we now turn our attention to its order properties. Recall that this was treated axiomatically (cf. Axioms (Oh ­ (O)J at the beginning of Chapter 2) by means of a subset P C JR called the subset of positive real numbers. In what follows we will define this subset and will denote it by JR+ , rather than P. Definition A.7 (Positive and Negative Real Numbers). We define a real number ~ E JR to be positive (resp ., negative), and write ~ > 0 (resp. , ~ < 0), if ~ = [x] for some x E C+ (resp., x E C-). The set of all positive (resp ., negative) real numbers will be denoted by JR+ (resp., JR-).

Proposition A.5. We have JR- = -JR+ := {~ E JR : -~ E JR+}, and the set JR+ of positive real numbers satisfies the following properties . (1) JR+ + JR+ C JR+, (2) JR+ . JR+ C JR+ , and (3) JR = JR+ U {O} U JR-, where the union is disjoint. (trichotomy) Exercise A.5. Prove Proposition A.5. Now that the existence of the set JR+ of positive real numbers has been estab­ lished and that, in view of Proposition A.5, the order axioms (01), (02), and (03) are satisfied, all the order properties of the set JR of real numbers can be proved as before. For instance, given~ , TJ E JR, we write ~ ~ TJ to mean TJ - ~ E JR+ U {O} and the set JR is then totally ordered by the ordering ~ . Remarks. (1) We have defined the notion of Cauchy sequence once for (ax­ iomatically defined) real numbers in Chapter 2 and again, in this appendix, for mtional numbers (which are real numbers), using exclusively mtional e > O. To show that, for rational sequences, the two definitions are identical, we need only show the following: (Vg E JR+)(3g' E Q)(O < e' ~ g). This, however, follows at once from Proposition A.3. (2) Since the set JR we have constructed satisfies all the algebmic and order properties treated axiomatically in Chapter 2, the notion of convergent sequence can be defined as before. In other words , a sequence (~n) E JRN of real numbers converges to the limit>. E JR (in symbols lim(~n) = >.), if the following holds:

Our construction of real numbers was motivated by the intuitive idea that a real number should be the limit of a convergent sequence of rationals. The following proposition shows that this is indeed the case. Proposition A.6. Let ~ be a real number. For a sequence x E C to be a repre­ sentative of e, it is necessary and sufficient that lim(xn) = ~ . Proof. Suppose that ~ = [x], and let e E JR+ be given. Then, we can find g' E Q+ with g' ~ g . We can also find N E N such that

m, n;::: N => -g' < X m - X n < e', 536 Appendix A. Construction of Real Numbers

Given m ~ N, the real number X m -~ is the class of the sequence (xm -Xl,Xm­ X2," ') which, using (*), can be replaced by an equivalent one, (Yn) E C, such that Xm-~ = [(Yn)] and -e' < Yn < e' Vn E N. Therefore, -e' < xm-~ < e', and hence IXm - ~I < e. This shows that we have lim(xn) = ~. Conversely, suppose that lim(xn) = ~ and that ~ = [(Yn)] for a sequence (Yn) E C. Then, as we just proved, lim(Yn) = ~. It then follows that (xn) '" (Yn) (why?), and we have ~ = [(xn )]. 0 All the algebraic and order properties we have proved for the set lR := CIN are also shared by its subfield Q of rational numbers. We are finally ready to prove the completeness of lR which, in the axiomatic treatment, was called the Supremum Properly or Completeness Axiom. This property is not satisfied by the subfield Q. Since the Supremum Property is equivalent to Cauchy's Criterion (as was pointed out in the remark following the proof of Theorem 2.2.5), all we need is to prove this criterion for our set lR := CIN. Theorem A.2 (Cauchy's Criterion). A sequence (~n) E lRN is convergent if and only if it is a Cauchy sequence. Proof. The necessity of the condition is obvious, as we saw in the proof of Theorem 2.2.5. To prove the sufficiency, note that, by Proposition A.6, for each n E N, we can find a rational number X n E Q (recall that X n = [(xn,xn, ... )]) such that I~n - xnl < lin. Now

(Ve E lR+)(3N E N)(m, n ~ N => I~m - ~nl < eI3). Thus, ifm, n ~ max{N,3Ie}, then

IXm- xnl ~ IXm - ~ml + I~m - ~nl + I~n - xnl I e Icc e < m + 3 + ~ ~ 3 + 3 + 3 = e, and hence (x n ) E C. Let ~ = [(xn )]. We then have lim(xn) ~ and, since lim(~n - x n ) = 0, we get lim(~n) =~. 0 References

[AG96] Malcolm Adams and Victor Guillemin. Measure Theory and Probability. Birkhauser, Boston, MA, 1996. [AB90] Charalambos D. Aliprantis and Owen Burkinshaw. Principles of Real Analysis. Academic Press, New York, second edition, 1990. [Apo74] Tom Apostol. Mathematical Analysis. Addison-Wesley Publishing Co., Inc. , Reading, MA, second edition, 1974. [Ash72] Robert B. Ash. Real Analysis and Probability. Academ ic Press, Inc., NY,1972. [ABu66] E. Asplund and L. Bungart. A First Course in Integmtion. Holt, Rine­ hart and Winston, New York, 1966. [BSOO] R. G. Bartle and D. R. Sherbert. Introduction to Real Analysis. John Wiley & Sons, New York, third edition, 2000. [Bea97] Alan F . Beardon. Limits. A New Approach to Real Analysis. Springer­ Verlag, New York, 1997. [Ber94] Sterling K. Berberian. A First Course in Real Analysis. Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1994. [BW90] Piotr Biler and Alfred Witkowski. Problems in Mathematical Analysis. Marcel Dekker, Inc., New York" 1990. [Bil95] Patrick Billingsley. Probability and Measure (Third Edition). John Wi­ ley & Sons Inc., New York, 1995. [Bin77] K. G. Binmore. Mathematical Analysis, a Stmightforward Approach. Cambridge University Press, Cambridge, 1977. [BM77] G. Birkhoff and S. MacLane. A Survey of Modern AIgebm. Macmillan Publishing Co., New York, fourth edition, 1977. 538 References

[Boa60] R. P. Boas. APrimer of Real Functions. Carus Mathematical Mono­ graphs No. 13, John Wiley & Sons, New York, 1960. [Bou74] N. Bourbaki. Elements of Mathematics; Theory of Sets. Addison­ Wesley Publishing Co., Reading, MA, 1974. [Bou68] N. Bourbaki. General Topology. Addison-Wesley Publishing Co., Read­ ing, MA, 1968. [Bri98] Douglas S. Bridges. Foundations of Real and Abstmct Analysis, An Introduction. Springer-Verlag, New York, 1998. [Br096] Andrew Browder. Mathematical Analysis, An Introduction. Springer­ Verlag, New York, 1996. [BP95] Arlen Brown and Carl Pearcy. An Introduction to Analysis. Springer­ Verlag, New York, 1995. [BB70] J. C. Burkill and H. Burkill. A Second Course in Mathematical Analysis. Cambridge University Press, Cambridge, 1970. [BK69] C.W. Burrill and J . R. Knudsen Real Variables. Holt, Rinehart and Winston, New York, 1969. [CK99] M. Capiriski and E. Kopp. Measure, Integml and Probability. Springer­ Verlag, London, 1999. [CZ01] M. Capiriski and T. Zastawniak. Probability Through Problems . Springer-Verlag, New York, 2001. [CarOO] N. L. Carothers. Real Analysis. Cambridge University Press, Cam­ br idge, 2000. [Cha95] Soo Bong Chae. Lebesgue Integmtion. Springer-Verlag, New York, sec­ ond edition, 1995. [Ch066] Gustave Choquet . Topology. Academic Press Inc., New York, 1966. Translated by Amiel Feinstein. [Chu74] Kai Lai Chung. Elementary Probability Theory with Stochastic Processes. Springer-Verlag, New York-Heidelberg-Berlin , 1974. [Co066] R. Cooper. Functions of Real Variables. D. Van Nostrand, London, 1966. [DM90] L. Debnath and P. Mikusiriski. Introduction to Hilbert Spaces with Ap­ plications. Academic Press, New York, 1990. [DS88] J. D. Depree and C. W. Swartz. Introduction to Real Analysis. John Wiley & Sons, New York, 1988. [Die69] Jean Dieudonne, Foundations of Modern Analysis. Academic Press Inc., New York, 1969. [DSc58] N. Dunford and J.T. Schwartz. Linear Operators. Interscience Pub­ lishers, New York, 1958. [Edg90] Gerald A. Edgar Measure, Topology, and Fractal Geometry. Springer­ Verlag, New York, 1990. [Edg98] Gerald A. Edgar Integml, Probability, and Fractal Measure. Springer­ Verlag , New York, 1998. References 539

[ER92] L. C. Evans and R. F. Gariepy. Measure Theory and Fine Properties of Functions. CRC Press, Inc. , 1992. [FaI85] K. J. Falconer The Geometry of Fractal Sets. Cambridge University Press, Cambridge, 1985. [FeI68] William Feller. An Introduction to Probability Theory and its Applica­ tions. John Wiley & Sons Inc. , New York, third edition, 1968. [Fis83] Emanuel Fischer. Intermediate Real Analysis. Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1983 [Fla76] Leopold Flatto. Advanced Calculus. The Williams & Wilkins Co., Bal­ timore, 1976. [FoI84] G. B. Folland. Real Analysis: Modem Techniques and Their Applica­ tions. John Wiley & Sons, New York, 1984. [Fri71] Avner Friedman. Advanced Calculus. Holt, Rinehart and Winston, New York, 1971. [Gal88] Janos Galambos. Advanced Probability Theory. Marcel Dekker Inc., New York, 1988. [Gar68] J . Garsoux. Analyse Mathematique. Dunod, Paris, 1968. [GeI92] B. R. Gelbaum. Problems in Real and Complex Analysis. Springer­ Verlag, New York, 1992. [Gha96] Saeed Ghahramani. Fundamentals of Probability. Prentice Hall, N. J., 1996. [Gof67] Casper Goffman. Real Functions. Prindle, Weber, and Schmidt, Boston, revised edition, 1967. [GoI64] R. R. Goldberg. Methods of Real Analysis. Blaisdell Publishing Co., New York, 1964. [Gor94] R. A. Gordon. The Integrals of Lebesgue, Denjoy, Perron, and Henstock. Grad. Studies in Math., vol. 4, Amer . Math. Soc., Providence, 1994. [HaI58] Paul Halmos . Finite Dimensional Vector Spaces. Van Nostrand, Princeton NJ , 1958; reprinted as Undergraduate Texts in Mathemat­ ics, Springer-Verlag, NY, 1974. [HaI50] Paul Halmos. Measure Theory. Van Nostrand, Princeton, NJ , 1950j reprinted as Graduate Texts in Mathematics, Springer-Verlag, NY 1975 [HaI60] Paul Halmos . Naive Set Theory. Van Nostrand, Princeton NJ , 1960j reprinted as Undergraduate Texts in Mathematics, Springer-Verlag, NY, 1974. [Her75] I. N. Herstein. Topics in Algebra. John Wiley & Sons, New York, second edition, 1975. [HS69] Edwin Hewitt and Karl Stromberg. Real and Abstract Analysis. Springer-Velag, Berlin-Heidelberg-New York, 1969. [KNOO] W. J . Kaczor and M. T. Nowak. Problems in Mathematical Analysis 1. Real Sequences and Series. Student Mathematical Library, vol. 4, Amer. Math. Soc., 2000 540 References

[KeI55] J. L. Kelley. General Topology. Van Nostrand, NY, 1955. [KG82) A. A. Kirillov and A. D. Gvishiani. Theorems and Problems in Func­ tional Analysis. Springer-Verlag, New York 1982. Translated from the 1979 Russian original by Harold H. McFadden. [Kn063) Konrad Knopp. Theory and Application of Infin ite Series. Blackie & Son Ltd., London, 1963. [K75] A. N. Kolmogorov. Foundations of the Theory of Probability. Chelsea Publishing Co., New York, second English edition, 1975. Translated from Russian by Nathan Morrison. [KF75) A. N. Kolmogorov and S. Fomin. Introductory Real Analysis. Dover Publications, Inc., NY 1975. Translated from Russian by Richard A. Silverman. [Kos95] Witold A. J . Kosmala. Introductory Mathematical Analysis. Wm. C. Brown Publishers, Dubuque, lA, 1995. [Kra91] Steven G. Krantz. Real Analysis and Foundations. CRC Press, Inc. , Boca Raton, FL, 1991. [Kre78] Erwin Kreyszig. Introductory Functional Analysis with Applications. John Wiley & Sons, NY, 1978. [Lan51] Edmund Landau. Foundations of Analysis. Chelsea, New York, 1951. [Lang69] Serge Lang. Real Analysis. Addison-Wesley Publishing Co., Reading, MA,1969. [MW99] J. N. McDonald and N. A. Weiss. A Course in Real Analysis. Academic Press, New York, 1999. [Mal95] Paul Malliavin. Integration and Probability. Springer-Verlag, NY, 1995. [Meg98] R. E. Megginson. An Introduction to Banach Space Theory. Springer­ Verlag, New York, 1998. [Mun75] J. R. Munkres. Topology, a First Course. Prentice Hall, Inc., NJ, 1975. [Munr71] M. E. Munroe. Measure and Integration. Addison-Wesley, Reading, MA, second edition, 1971. [Nev65] Jacques Neveu. Mathematical Foundations of the Calculus of Probabil­ ity. Holden-Day Inc., San Francisco, Calif., 1965. Translated by Amiel Feinstein. [Olm59] John M. H. Olmstead. Real Variables. Appleton-Century-Crofts, Inc., New York, 1959. [PedOO] Michael Pedersen. Functional Analysis in Applied Mathematics and En­ gineering. Chapman & Hall/CRC, NY/ Boca Raton, FL ., 2000. [Pedr94] George Pedrick. A First Course in Analysis. Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1994. [Per64] William J. Pervin. Foundations of General Topology. Academic Press Inc., New York, 1964. [PSz72] George P6lya and Gabor Szego, Problems and Theorems in Analysis. Springer-Verlag, Berlin-Heidelberg, vols. 1 & 2, 1972. References 541

[PR69] Yu. V. Prohorov and Yu. A. Rozanov. Probability Theory. Springer­ Verlag, New York-Heidelberg-Berlin, 1969. [Pr098] Murray H. Protter. Basic Elements of Real Analysis. Springer-Verlag, New York, 1998. [Ran68] John F. Randolph. Basic Real and Abstract Analysis. Academic Press, New York, 1968. [Ree98] Michael C. Reed. Fundamental Ideas of Analysis. John Wiley & Sons, New York, 1998. [RS72] Michael Reed and Barry Simon. Methods ofMathematical Physics. Aca­ demic Press, New York, vol. 1, 1972. [RN56] F. Riesz and B. Sz.-Nagy. Functional Analysis. Ungar, New York, (English edition), 1956. [Ros80] Kenneth A. Ross. Elementary Analysis: The Theory of Calculus. Springer-Verlag, New York, 1980. [Ros76] Sheldon Ross. A First Course in Probability. Macmillan, New York, 1976. [Roy88] H. L. Royden. Real Analysis. Macmillan, New York, third edition, 1988. [Rud74] Walter Rudin. Real and Complex Analysis. McGraw-Hill Book Co., New York, second edit ion, 1974. [Rud64] Walter Rudin. Principles of Mathematical Analysis. McGraw-Hill Book Co., New York, second edition, 1964. [Rud73] Walter Rudin. Functional Analysis. McGraw-Hill Book Co., New York, 1973. [Shi84] A. N. Shiryayev. Probability. Springer-Verlag, New York-Berlin­ Heidelberg, 1984. Translated from Russian by R. P. Boas. [Sch96] Michael J. Schramm. Introduction to Real Analysis. Prentice Hall, Inc., NJ,1996. [Spi94] Michael Spivak. Calculus. Publish or Perish Inc., Houston, TX, third edition, 1994. [StoOl] Manfred Stoll. Introduction to Real Analysis. Addison-Wesley Long­ man, Inc., second edition, 2001. [Sup60] P. C. Suppes. Axiomatic Set Theory. Van Nostrand, NY, 1960. [Tay96] Michael E. Taylor . Partial Differential Equations, Basic Theory. Springer-Verlag, New York, 1996. [Tit39] E. C. Titchmarsh. The Theory of Functions. Oxford University Press, second edition, 1939. [Tor88] Alberto Torchinsky. Real Variables. Addison-Wesley Publishing Co., Inc., Reading, MA, 1988. [VS82] A. C. M. Van Rooij and W. H. Schikhof A Second Course on Real Functions. Cambridge University Press, Cambridge, 1982. [VarOl] S. R. S. Varadhan Probability Theory. A.M.S. Courant Lecture Notes, vol. 7, Amer . Math. Society, Providence, Rl, 2001. 542 References

[WadOO) W. R. Wade. An Introduction to Analysis. Prentice-Hall, Inc., NJ, 2000. [Wei73) Alan J. Weir. Lebesgue Integmtion and Measure. Cambridge University Press, Cambridge, 1973. [Yos74) K6saku Yosida. Functional Analysis. Springer-Verlag, Berlin- Heidelberg-New York, fourth edition, 1974. Index

Abel's partial summation formula, almost uniform convergence, 444 63 alternating series , 64 Abel's Test, 63 angular point, 211 Abel's Theorem, 65, 320 antiderivative (primitive), 279 Abelian (commutative), 17 Appolonius 's identity, 390 absolute value , 40, 298 approximation (uniform), 167 absolutely continuous, 154, 295 Archimedian Property, 43 absolutely convergent series, 60 arcwise connected, 199 absolutely summable, 71 area under the graph, 254, 522 accumulation point, 48, 162 Arzela-Ascoli Theorem, 381 additive (function) , 124 associativity, 16, 73 adjoint operator, 391 asymptote aleph nought (No), 29 horizontal, 98 algebra, 21 vertical, 96 (7-, 4 at random, 497 Banach,357 atom, 514 Borel,435 Average Value, 289 commutative, 21 Axiom of Choice, 9, 28 division, 21 normed,352 B(n,p),501 sub-,21 Baire Category Theorem, 169 algebra of sets, 4 Baire metric, 203 almost all (a.a.), 265 ball almost elementary, 448 closed , 161 almost everywhere (a.e.), 265, 469 open, 161 convergence, 402 Banach almost surely (a.s.}, 497 the space Co of, 358 544 Index

Banach algebra, 357 function, 16 Banach space, 357 pointwise, 382 Banach's Fixed Point Theorem, 181 uniformly, 70, 382 Banach-Steinhaus Theorem, 363 bounded away from zero, 107 base Bounded Convergence Theorem, 518 countable, 165 bounded functions Basic Counting Principle, 23 metric space of, 167 basis, 20 bounded set, 9, 160, 355 orthogonal, orthonormal, 372 bounded variation, 284 Schauder, 387 bounded, unbounded (sequence), 49 Bayes's formula, 498 Beppo Levi's Theorem, 413 cr, Coo, 231 Bernoulli Calderon's proof numbers, 348 of Steinhaus's Theorem, 465 polynomials, 348 canonical projection, 13 random variable, 501 canonical representation, 441 sequence, 496 Cantor set, 118, 119 trial,496 generalized, 266 Bernoulli's inequality, 40, 224 Hausdorff dimension of, 474 Bernstein Approximation Theorem, measure of, 266 148, 512 Cantor's diagonal method, 380 Bernstein polynomials, 148 Cantor's ternary function, 122, 125 Bessel functions, 345 Cantor's Theorem, 32, 168 Bessel's inequality, 331, 372, 375 Cantor-Bendixon Theorem, 169 Best Approximation, 331, 374 Caratheodory's definition, 455, 474 big 0 ,106 Caratheodory's Theorem, 212 binary expansion, 46 cardinal number (or cardinality), 29 binary operation, 16 Cartesian product, 6, 28 binomial coefficients, 23, 24, 319 Cauchy in measure, 515 Binomial Formula, 24 Cauchy product, 64, 75 binomial random variable, 501 Abel's Theorem on, 321 Birkhoff and MacLane, 16 Cauchy sequence, 54, 167 Bisection Method, 135 negative, 534 Bolzano-Weierstrass Property, 188 positive, 534 Bolzano-Weierstrass Theorem, 53 Cauchy's Condensation Theorem, 59 Borel algebra, 435 Cauchy's Criterion, 54, 56, 71, 72, completion of, 450 93, 97, 300, 374, 375, 398, Borel function, 462, 48J. 536 Borel set, 435 uniform, 304 Borel-Cantelli Lemma Cauchy's functional equation, 113, First, 499 124,446 Second,500 Cauchy's inequality, 40, 301 bound Cauchy-Hadamard Theorem, 313 least upper, greatest lower, 9 Cauchy-Schwarz inequality, 67, 276, upper, lower, 9 329,367,458,519 boundary, 163 chain (totally ordered set), 9 point, 163 chain connected, 207 bounded Chain Rule , 218 above, below, 9 Change of Variables, 281, 505, 507 Index 545 characteristic function, 15 content zero (set of) , 259 of a random variable, 526 continuity characterization of intervals, 44 at a point, on a set, 172 Chebyshev's inequality, 463, 485, 509, global definition of, 174 528 sequential definition of, 124, 173 choice function, 28 continuous, 122, 172 class, 7 Continuous Extension Theorem, 140 equivalence, 7 continuous extensions, 182 representative of a, 7 continuum (c), 29 class en (function of), 231 contraction (mapping), 141, 180 closed ball, 161 contractive map, 154, 206 Closed Graph Theorem, 192, 364 contractive sequence, 55 closed set , 47, 162 convergence closure , 88 absolute, 60, 398 relative, 165 almost everywhere, 402 closure (of a set) , 163 almost surely, 510 cluster point, 88, 163 almost uniform, 444 coin-tossing, 496 conditional, 60, 398 commutative ring , 18 in measure, 463, 515 compact, 115, 186 in probability, 510 countably, 188 interval of, 312 Frechet, 188 normal,306 relatively, 187 of a sequence, 47 sequentially, 188 of Fourier series , 336 compactness, 187 of series, 55 complement, 2 pointwise, 301 complement (of a subspace), 20 radius of, 312 complete, 167 uniform, 303 Completeness Axiom (Supremum Prop­ convergent, divergent, 48 erty),42 convergent, divergent (series) , 55 completion, 184, 390 convex function, 240 of a normed space, 358 convex hull, 386 of Borel algebra, 450 convex set, 206, 368 complex conjugate, 300 convolution of Borel measures, 521 complex number, 297 convolution of functions, 520 composite function, 14 correlation coefficient , 526 composition , 6 cosine function, 325 of relations, 6 countable additivity, 437 concave function, 240 countable base, 165 condensation point, 169, 202 countable set, 29 conditional probability, 498 countable subadditivity, 437 conditionally convergent series, 60 countably compact, 188 congruence modulo n, 7 countably infinite, 29 conjugate linear, 367 covariance, 525 connected (metric space), 194 cover arcwise, 199 locally finite, 205 locally, 197 open, 115, 186 connected component, 197 Criterion connected, disconnected, 117 Cauchy's, 54 546 Index

Dini's, 336 Dirichlet's integral, 335 Lebesgue's Integrability, 270 Dirichlet's kernel, 332 Lusin's, 449 Dirichlet's Test, 63, 425 Uniform, 343 Darboux integrals, 253 Dirichlet's Theorem, 77 Darboux sum, 252 discontinuity Darboux's Theorem, 222, 256 infinite, 128 De Morgan's Laws, 3 jump, 128 decimal expansion, 46 of the first kind, 129 decreasing, 86 of the second kind, 129 dense, 44, 165 removable, 128 nowhere, 118 discontinuous, 122 density function, 503 discrete, 126 joint, 507 random variable, 501 density of Q (in R), 44 distance (metric), 158 derivative, 210 Hausdorff, 201 left, right, 211 in R, 47 Schwarzian, 247 transported, 178 symmetric, 246 distribution function, 473 derived set, 201 cumulative, 479 diagonal, 7, 159 of a random variable, 502 diameter, 160 divergent diffeomorphism, 245 sequence, 166 difference operator, 248 series, 55 difference set, 2 division algebra, 21 differentiable, 210 Division Algorithm, 26 n-times, 230 division ring, 18 n-times continuously, 231 domain, 6 infinitely, 231 Dominated Convergence Theorem, 489, uniformly, 246 518 Differential Calculus, 216 domination (set-), 30 differential equation double, multiple (sequence), 70 Legendre's, 248 double, multiple (series) , 70 differential operator, 238 du Bois-Raymond Test, 344 symbol of, 238 dual differentiation (algebraic), 354 term-by-term, 311 (topological), 356 under the integral sign, 428 dimension, 21 E[X),506 orthogonal, 377 s-neighborhood, 47 Dini's Criterion, 336 e (natural base), 58 Dini 's Theorem, 305, 306 irrationality of, 59 Dirac measure, 469 Edelstein's Theorem, 206 direct (or Cartesian) product, 6, 28 Egorov's Theorem, 443, 516 infinite, 28 element, 1 direct image , 13 maximal (minimal), 9 direct sum, 20 elementary functions, 214 directed set, 11, 470 derivatives of, 214 Dirichlet function, 125, 255 elementary set, 447, 491 Index 547 elementwise method, 2 favorable, 522 enumeration, 29 Fejer's integral, 335 envelope Fejer's kernel, 332 upper, lower, 123 Fejer's Theorem, 339 equicontinuous, 380 Fermat, Pierre de, 261 uniformly, 380 field, subfield, 19 equivalence class, 7 finite (real number), 47 equivalent (or equipotent, equipol- finite dimensional, 20 lent) sets, 29 Finite Intersection Property, 150, 187 equivalent functions, 104 finite set , 15 equivalent metrics, 162, 178 first category (meager), 165 equivalent norms, 352, 356 First Comparison Test, 57 Euclidean n-space, 28, 159 First Fundamental Theorem, 280 Euler's ¢-function, 27 Fixed Point Theorem, 134, 143 Euler's Beta Function, 292 Formula Euler's Constant, 78 Binomial, 24 Euler's equation, 325 Multinomial, 24 Euler's Theorem, 245 Taylor's , 236 event, 496 Fourier coefficient, 330, 374 events Fourier series , 330, 374 independent, 498 Fourier transform, 429 limsup, liminf of, 499 of a measure, 521 eventually, 499 Frechet compact, 188 expansion fractional powers (roots), 44 binary, 46 Fresnel integrals, 425 decimal,46 Fubini-Tonelli Theorem, 494 ternary, 46 function, 12 expansive map, 206 n-th iterate of, 206 expectation, 505 absolutely continuous, 154, 295 experiment, 495 absolutely summable, 71 exponential function additive, 124 complex, 322 Borel, 462, 481 derivative of, 215 bounded above, below, 16,86 general, 324 bounded, unbounded, 16, 86 real,323 Cantor's ternary, 122 extended real line, 47, 88, 178 characteristic, 15 Extension Theorem, 476 choice, 28 exterior, 163 complex exponential, 322 point, 163 composite, 14 extrema continuous, 122, 167, 172 global,221 contractive, 154, 206 local, 221, 249 convex, concave, 240 Extreme Value Theorem, 132, 191 differentiable (smooth), 210 Dirichlet, 125, 255 :F" , 453 discontinuous, 122 F. Riesz's Lemma, 362 distribution, 473 F . Riesz's Theorem, 421 domain, range of, 12 Falconer, 473 Euler's Beta, 292 Fatou's Lemma, 419, 487, 518 extended real-valued, 47 548 Index

Gamma, 399 total variation, 287 general exponential, 324 uniformly continuous, 137, 179 general power, 324 unordered sum of, 69 graph of, 175 with compact support, 422 greatest integer, 129 functions homogeneous, 245 equivalent, 104 identity, 14, 126 trigonometric, 325 increasing at a point, 110 increasing, decreasing, 86 96,453 integrable, 488 Gamma function, 399 inverse of, 14 Gauss's Test, 62 jump, 130 Geometric mean-Arithmetic mean In­ Lebesgue integrable, 408, 457 equality, 41 Lebesgue measurable, 432 Geometric mean-Arithmetic mean in­ left continuous, 127 equality, 249 limit of, 88, 170 geometric series, 56 linear, 124 ratio of, 56 Lipschitz, 140, 180, 225 global extrema, 221 maximum, minimum of, 86 Gram-Schmidt Orthogonalization, 373 measurable, 481 graph,175 monotone, 87 greatest common divisor (gcd), 26 natural logarithm, 323 greatest integer function, 129 nowhere differentiable, 311 greatest lower bound (inf), 9 of bounded variation, 284 Gronwall's inequality, 225 one-to-one (injective), 13 group, 16 onto (surjective), 13 Abelian (commutative), 17 oscillation of, 124, 203 symmetric, 17 per iodic, 125 piecewise cont inuous, 135, 260 Hahn-Banach Theorem, 365 piecewise linear, 144, 167, 260 Halmos, 16, 28, 436 piecewise monotone, 135 harmonic series , 56 piecewise smooth, 340 alternating, 60 polynomial, 167 Hausdorff dimension, 473 rational, 126 of the Cantor set, 474 real analytic, 317 Hausdorff distance, 201 real exponential, 323 Hausdorff measure, 477 regulated, 262 Hausdorff outer measure, 473 Riemann Zeta, 349 Hausdorff-Lennes separation condi- right continuous, 127 tion, 195 right differentiable, 211 Heine-Borel Theorem, 116 sawtooth, 311 Hellinger-Toeplitz Theorem, 392 simple, 441, 483 Herstein, 16 sine, cosine, 325 higher derivatives, 230 square integrable, 458 Hilbert space, 367, 391 step, 144, 167, 260, 400 Hilbert spaces subexponential, 247 L 2(E, IF), 458 summable, 69 L~(X,IF), 519 support of, 422 e2 (J, Jli'), 377 supremum, infimum of , 86 isomorphic, 377 Index 549

Holder's inequality, 243, 385, 518 Sobolev, 350 homeomorphic, 136, 177 Triangle, 40, 41, 351 homeomorphism, 136, 177, 192 ultrametric, 160 homogeneous, 245 Young's, 290 homomorphism, 22 Infimum Property, 42 Hormander's Generalized Leibniz Rule, infinite dimensional, 20 239 infinite limit, 52 horizontal asymptote, 98 infinite set , 15 horizontal slice, 193, 199 infinitely often, 499 hyperplane, 371 infinitesimal, 107 order of, 109 ideal, 19 principal part of, 109 maximal,34 infinity (±oo), 46 identity element, 17 initial segment, 10 identity function, 14 injective, 13 image (direct, inverse), 13 inner mesure (Lebesgue), 454 imaginary part, 300 inner product, 367 imaginary unit, 298 integers, 4 improper Riemann integral, 396 integrable function, 254, 488 Inclusion-Exclusion Principle, 24,436 absolutely, 255 increasing, 86 Lebesgue, 408 increasing at a point, 110 integral increasing, decreasing (sequence), 49 indefinite, 279 indefinite integral, 279 Lebesgue, 456 independent events, 498 linearity of , 488 independent families of events, 498 lower Darboux, 253 indeterminate forms, 103 Riemann, 254 index set, 28 upper Darboux, 253 Induction Integral Test (Cauchy's) , 289 Principle of Mathematical, 10 integration Principle of Strong, 11 by parts, 282 Principle of Transfinite, 11 by substitution, 281 inequality term-by-term, 308, 342 Bernoulli's, 40, 224 interior, 163 Bessel's, 331, 372, 375 point, 88, 163 Cauchy's, 40 relative, 165 Cauchy-Schwarz, 67, 276, 329, Intermediate Value Property, 133, 221 367, 458, 519 Intermediate Value Theorem, 133 Chebyshev's, 463, 485, 509, 528 intersection, 2 Geometric mean-Arithmetic mean, interval, 42 41,249 bounded, unbounded, 42 Gronwall's, 225 endpoint(s) of, 42 Holder's, 243, 385, 518 length of, 264 Jensen's, 240, 290 open, half-open, closed, 42 Kolmogorov's, 509 Interval Additivity Theorem, 271 Landau's, 248 interval of convergence, 312 Markov's, 509 inverse element, 17 Minkowski's, 68, 243, 367, 385, inverse function, 14 519 derivative of, 219 550 Index

Inverse Function Theorem, 137 t 2 , 67 inverse image, 13 Lagrange's identity, 299 irrationality of v'2, 44 Lagrange's remainder, 236 irrationality of e, 59 >. (Lebesgue measure), 434 isolated point, 48, 162 Landau's inequality, 248 isometric, 177 Landau's 0, 0, 106 isometric isomorphism, 356, 377 lattice, 11, 383 isometry, 177, 356 distributive, 12 isomorphic lattice identities, 11 algebras, 23 Law of Multiplication, 522 fields, 23 least upper bound (sup), 9 groups, 23 Lebesgue covering property, 205 rings, 23 Lebesgue inner measure, 454 vector spaces, 23 Lebesgue integrable (function), 408, isomorphic (topologically), 356 457, 488 isomorphism, 23 Lebesgue integral, 409, 423, 456, 488 isomorphism (topological), 356 in cs, 406 iterated sum, 73 of a nonnegative function, 484 Lebesgue measurable function, 432 Jensen's inequality, 240, 290 Lebesgue measurable set, 434 joint density, 507 Lebesgue measure, 434 joint distribution, 507 completeness of, 436 Jordan Decomposition Theorem, 287 regularity of, 449 Jordan, Camille, 283 Lebesgue outer measure, 451 jump (of a function), 128 Lebesgue sum, 456 jump function, 130 Lebesgue's Bounded Convergence The­ orem, 418 Kelley, John, 254 Lebesgue's Dominated Convergence kernel Theorem, 417 Dirichlet's, 332 Lebesgue's Integrability Criterion, 270 Fejer's, 332 Lebesgue's Monotone Convergence The­ Poisson, 350 orem, 414 kernel (null space), 356 Lebesgue-Stieltjes Kolmogorov, 495 measure, 477 Kolmogorov's inequality, 509 outer measure, 473 Kronecker's delta, 15 left continuous, 127 Kummer's Test, 62 left limit, right limit, 94 Legendre's differential equation, 248 cs, .e0(I), 432 Legendre's Polynomials, 247 c , .el(I), 408 Leibniz Rule, 231 .e2, L 2, 458 Horrnander's Generalized, 239 .et, .et(I), 405 Leibniz's Test, 64 L°(I), LO, 432 Uniform, 344 L l , 420 Lemma A (Riesz's), 402 L~(X), L~(X ,C) , 489 Lemma B (Riesz 's) , 403 Lt(X,IF), 518 length, 119 c (X,Y) , 481 Lerch's Theorem, 290 .c~(X ,lR) , .c~(X,C) , 488 L'H6pital's Rule, 226 l t , e. tOO , 200 limit, 48, 166 Index 551

infinite, 52, 96 map (or mapping), ~2 left , right, 94 (bounded) multilinear, 356 one-sided, 94 contraction, 180 properties of, 91 linear, 22, 23, 354 sequential definition of, 171 open, closed , 176 uniqueness of, 166 marginal distributions, 507 upper, lower, 53, 123 Markov's inequality, 509 limit (of a function), 88 maximal (minimal) element, 9 Limit Comparison Test, 57 maximal ideal , 34 limit point, 48, 88, 162 maximum, minimum, 9, 86 Limit Theorems, 51 global, 221 lim sup, lim inf, 53 local , 152, 221 Lindelof', 114 meager ( of first category), 165 Lindelof property, 189 mean, 505 Lindelof space, 189 mean square approximation, 341 linear, 124 Mean Value Theorem, 223 linear combination, 20 Cauchy's, 225 linear functional, 354 for Integrals (First), 277 linear map, 22, 23, 354 for Integrals (Second), 277 bounded, 356 Mean Value Theorem for Integrals, kernel of, 356 277, 427 linear operator, 354 measurable function, 481 linearly independent, 20 Lebesgue's defin ition of, 439 Lip, LipQ, 140, 180, 358 measurable set Lipschitz, 140, 180, 225, 358 Lebesgue's defin ition of, 454 condition, 140 measurable space, 468 constant, 140, 180 measure, 467 locally, 141, 181 a-finite , 467 of order o , 141 complete, 436, 469 little 0 , 106 completion of, 469 Littlewood's Theorem, 249 continuity of, 468 local extrema, 221, 249 convergence in, 463, 515 local homeomorphism, 204 countable subadditivity of, 468 locally bounded, 204 counting, 469 locally closed, 204 Dirac, 469 locally compact, 205 finite, 434, 467 locally connected, 197 finite additivity of, 468 locally finite, 164, 205 finite subadditivity of, 468 locally Lipschitz, 141 Fourier transform of, 521 of order o, 141 Hausdorff, 477 locally open, 204 Lebesgue, 434 Location of Zeroes Theorem, 134 Lebesgue-Stieltjes,477 logarithm (natural), 92 metric outer, 479 Lusin's Criterion, 449 monotonicity of, 468 Lusin's Theorem, 444 outer, 472 probability, 496 M.x(IR), Mn(IR), 434 product, 493 m-tail, 48, 166 measure space, 468 Maclaurin series, 318 measure zero , 265, 435, 469 552 Index

Mertens' Theorem, 64 Nested Intervals Theorem, 45 mesh (or norm), 257 Newton's Binomial Theorem, 320 metric Newton-Raphson process, 248 associated with a norm, 352 non-atomic, 514 Baire, 203 nonmeasurable, 438 discrete, 158 norm, 351 product, 159 L 1_, 421 uniform, 158, 167 L 2_, 391 metric (distance), 158 LP-, 519 1 metric outer measure, 479 e _, 353 metric property, 204 e2_, 67, 353 metric space, 158 e-, 385 chain connected, 207 Euclidean, 352 complete, 167 sup-, 353 completion of, 184 norm (or mesh), 257 connected, 194 normed algebra, 352 countably compact, 188 normed space, 352 Frechet compact, 188 finite dimensional, 359 locally compact, 205 quotient, 361 product, 159 separable, 362 second countable, 165 nowhere dense, 118, 165 separable, 165 nowhere differentiable, 311 sequentially compact, 188 null sequence, 52, 532 metrics null set, 469 equivalent, 178 null space (kernel), 356 uniformly equivalent, 205 numbers middle third (open), 118 complex, 4, 297 Minkowski's inequality, 68, 243, 367, extended, 47 385, 519 irrational, 44 module, 20 natural,3 moments (of a random variable), 527 prime, 27 monotone (function), 87 rational,4 monotone class , 492 real, 4, 37 Monotone Convergence Theorem, 50, 486, 518 one-sided limits, 94 Monotone Limit Theorem, 94, 101 infinite, 96 monotone sequence, 49 one-to-one, 13 of sets, 491 one-to-one correspondence (bijective), monotonicity, 436 13 Monte Carlo method, 528 onto, 13 multilinear map, 356 open, 114 Multinomial Formula, 24 locally, 204 MVT,223 open ball, 161 open cover, 115, 186 n-space open interval, 42 Euclidean, 28 in an, 472 Unitary, 28 open map, 176 natural logarithm, 323 Open Mapping Theorem, 364 derivative of, 220 open set, 47, 162 Index 553 operation (binary), 16 parallelogram law, 368 operator Parseval's adjoint, 391 Identity, 376 bounded, 356 Relation, 342, 375, 460 difference, 248 Theorem, 341, 460 differential, 238 partial ordering, 8 self-adjoint, 392 partial sum (of a function), 69 shift, 392 partial sum (of series), 55 ordered partition, 7, 194, 251 n-tuple,6 mesh (or norm) of, 257 pair, 6 refinement of, 252 linearly, 9 tagged,252 partially, 8 path component, 199 totally, 9 path connected, 199 well, 10 peak, 50 ordering perfect set, 49, 162, 169 lexicographic (or dictionary), 12 period, 125 partial,8 periodic function, 125 total,9 continuous, 126 well, 10 permutation, 13 ordinate set, 522 permutation, combination, 23 orthogonal piecewise continuous function, 135, basis, 372 260 complement, 370 Piecewise Linear Approximation, 145 projection, 370 piecewise linear function, 144, 260 system, 330 piecewise monotone function, 135 vectors, 369 piecewise smooth function, 340 orthogonal dimension, 377 point orthogonal system, 372 accumulation, 48 complete, 372 angular, 211 orthogonalization condensation, 169, 202 Gram-Schmidt, 373 isolated, 48 orthonormal limit, 48 basis, 372 pointwise orthonormal system, 372 convergence, 301 complete, 372 limit, 301 oscillation, 124 Poisson kernel, 350 at a point, 203, 268 polarization identity, 390 on a set, 203, 268 Polya-Szego, 23 Osgood's Theorem, 184, 363 Polynomials outer measure, 472 Legendre's, 247 Hausdorff, 473 Taylor, 234 Lebesgue, 451 power function (general), 324 Lebesgue (on an) , 472 Power Rule, 214, 261 Lebesgue-Stieltjes, 473 General, 218 metric, 479 power series, 312 power set, 2 11", 326 pre-Hilbert space, 367 p-series, 56 premeasure, 475 554 Index prime random variable, 500 factorization, 27 absolutely continuous, 503 number, 27 Bernoulli, 501 primitive (antiderivative), 279 binomial, 501 Principle of Analytic Continuation, Cantor-Lebesgue, 503 347 Cauchy, 504 Principle of Isolated Zeroes, 347 characteristic function of, 526 Principle of Mathematical Induction, constant, 501 10 continuous, 502 Principle of Strong Induction, 11 density function of, 503 Principle of Transfinite Induction, 11 discrete, 501 probability, 497 distribution function of, 502 classical, 497 expectation of, 505 conditional, 498 exponential, 504 probability distribution, 500 geometric, 524 probability measure, 496 mean of, 505 probability space, 496 negative binomial, 524 product normal (or Gaussian), 504 Cauchy, 64 Poisson, 501 direct (or Cartesian), 6, 28 probability distribution of, 500 product a-algebra, 490 simple, 501 product (metric) space, 159 square-integrable, 506 complete, 170 standard deviation of, 506 convergence in, 170 standard normal, 504 product measure, 493 uniform, 504 product metric, 159 uniformly distributed, 502 Product Rule, 216 variance of, 506 projection random variables canonical, 13 identically distributed, 504 orthogonal, 370 independent, 504 proper inclusion, 1 jointly continuous, 507 pseudometric, 200 uncorrelated, 525 pseudometric space, 200 range, 6 pullback by a surjection, 423 rate of change Pythagorean Theorem, 371 average, 221 instantaneous, 221 quantifier, 3 Ratio Test, 61 existential, 3 Uniform, 343 universal, 3 rational function, 126 quaternions (real), 21 real analytic function, 317 Quotient Rule, 216 real numbers, 37 quotient set, 7 addition of, 532 quotient space, 361 construction of, 531 multiplication of, 532 Raabe's Test, 62 subtraction of, 532 Rademacher functions, 529 real part, 300 radius of convergence, 312 rearrangement, 65 random, 496 rectangle, 490 random selection, 497 rectifiable curve, 295 Index 555 reflexive space, 389 Rolle's Theorem, 222 regularity Root Test, 60 of Lebesgue measure, 449 Rudin, 487, 494 regulated function, 262 relation, 6 sample point, 497 composite, 6 sample space, 496 domain of, 6 sawtooth function, 311 equivalence, 7 scalar multiplication, 19 extension of, 6 Schauder basis, 387 inverse of , 6 Schroder-Bernstein Theorem, 31 range of, 6 Schwarzian derivative, 247 reflexive, 7 second category, 165 restriction of, 6 Second Comparison Test, 59 symmetric, 7 second countable, 165 transitive, 7 second dual, 389 relative interior, closure, 165 Second Fundamental Theorem, 280 relative topology, 163 self-adjoint operator, 392 relatively open, closed, 162 seminorm, 352 relatively prime, 26 seminormed space, 352 remainder separable, 165 Cauchy's form of, 237 separating points, 383 Lagrange's, 236 separation, 195 repeated sum, 73 sequence, 13 representative, 7 m-tail of, 48 resolvent, 389 bounded, unbounded, 49 equation, 390 Cauchy, 54, 167 set , 389 contractive, 55 Riemann integrable, 254 convergence of, 47 Riemann integral, 254 convergent, divergent, 48, 166 improper, 396 double, multiple, 70 Riemann sum, 252 increasing, decreasing, 49 Riemann Zeta Function, 349 limit of, 48, 166 Riemann's Lemma, 256 monotone, 49, 402 Riemann's Localization Theorem, 336 null , 52 Riemann's Theorem (on rearrange- pointwise convergent, 301 ments),66 strictly increasing, decreasing, Riemann-Darboux Theorem, 255 49 Riemann-Lebesgue Lemma, 332, 423 uniformly convergent, 303 Riesz integrable, 457 sequential definition Riesz Representation Theorem, 371 of continuity, 124, 173 Riesz-Fischer Theorem, 376 of limit, 171 right continuous, 127 sequentially compact, 188 right differentiable, 211 series, 55 ring, 18 Abel's Test, 63 u-,4 absolutely convergent, 60, 360 commutative, 18 alternating, 64 division, 18 alternating harmonic, 60 with unit element, 18 Cauchy product, 64 ring of sets, 4 conditionally convergent, 60 556 Index

convergent, divergent, 55, 360 finite, infinite, 15 Dirichlet's Test, 63 interior of, 163 double, multiple, 70 interior point, interior (of), 88 First Comparison Test, 57 Lebesgue measurable, 434 Fourier, 330 linearly ordered, 9 Gauss's Test, 62 nonmeasurable, 438 geometric, 56 nowhere dense, 118, 165 harmonic, 56 null (or of measure zero), 469 Kummer's Test, 62 of first category, 165 Leibniz's Test, 64 of measure zero, 265 Limit Comparison Test, 57 of second category, 165 Maclaurin, 318 open, closed , 47, 162 normally convergent, 306 partially ordered, 8 p-,56 partition of, 7 partial sum of, 55 perfect, 49, 162 pointwise convergent, 306 quotient, 7 power, 312 relatively compact, 187 Raabe's Test, 62 totally bounded, 189 Ratio Test, 61 totally disconnected, 117 rearrangement of, 65 totally ordered, 9 Riemann's Theorem, 66 uncountable, 29 Root Test, 60 universal, 1 Second Comparison Test, 59 sets, 2 square summable, 67 algebra of, 4 Taylor, 318 disjoint, 2 trigonometric, 330 equivalent, 29 uniformly convergent, 306 pairwise disjoint, 7 unordered, 69 ring of, 4 sesquilinear form, 367 shift operator, 392 set, 1 o-algebra, 4 :Fu , 201 product, 490 (h, 201 u-finite measure, 467 Borel,435 u-ring,4 boundary of, 163 simple function, 441, 483 bounded, 9, 160, 355 canonical representation of, 441 Cantor, 118 integral of, 484 closure of, 88, 163 sine function, 325 compact, 115, 186 slice connected, disconnected, 117, 194 horizontal, 199 convex , 206, 368 vertical, 199 countable, 29 smooth,210 countably infinite, 29 smoothness of inverse functions, 219 dense, 165 Sobolev inequality, 350 derived, 201 space diameter of, 160 n-dimensional Euclidean, 159 directed, 11 Euclidean, 159 discrete, 126 measurable, 468 elementary, 447, 491 measure, 468 exterior of, 163 metric, 158 Index 557

normed,352 proper, 1 probability, 496 subspace pseudometric, 200 metric, 158 sample, 496 Substitution Theorem, 316 seminormed, 352 sufficiently close, 89, 98 Washington D. C., 200 sufficiently large, 98 spaces sum 8(X), 8(X, Y), 356 Darboux, 252 £(X), £(X, Y), 354 iterated, 73 £(X,lF),354 Lebesgue, 456 L1(X), L1(X, c), 489 repeated, 73 L (X , lF), 518 Riemann, 252 i:C(X, Y), 481 unordered, 69 .c1(X, R) , .c1(X, C), 488 summable, 374 r-, f1, f2 , 353 absolutely, 71, 374 f P-, 385 summable function, 69 Lip, LipQ, 358 Suppes, 11 co,358 support, 422 Banach,357 compact, 422 Hilbert, 367, 391 support line, 243 pre-Hilbert, 367 Supremum Property (Completeness span, 20 Axiom),42 spectrum, 389 supremum, infimum, 9, 86 sphere, 161 surjective, 13 unit, 194 symbol,238 square integrable function, 458 symmetric derivative, 246 square root (existence of), 43 symmetric difference, 2 square summable series, 67 symmetric group, 17 Squeeze Theorem, 91, 100 in £1, 419 tag, 252 in M.x(R), 436 Tauber's Theorem, 347 standard deviation, 506 Taylor coefficients, 234 Steinhaus's Theorem, 445 Taylor Polynomials, 234 Calderon's proof of, 465 Taylor series, 318 step function, 144, 260, 400 Taylor's Formula integral of, 273, 400 with integral remainder, 283 Step Function Approximation, 144 Taylor's Formula with Lagrange's Re­ Stone-Weierstrass Theorem, 383 mainder, 236 Complex, 385 Taylor's Theorem, 317 strictly increasing, decreasing (sequence), term 49 n-th, 13 Strong Law of Large Numbers, 511 ternary expansion, 46, 121 subcover (open), 115 ternary set subexponential function, 247 Cantor's, 118 subgroup, 17 Test subring,19 Abel's, 63 subsequence, 50 Dirichlet's, 63 monotone, 50 First Comparison, 57 subset, 1 Gauss's, 62 558 Index

Kummer's, 62 Lebesgue's Bounded Convergence, Leibniz's, 64 418 Limit Comparison, 57 Lebesgue's Dominated Conver­ Raabe's, 62 gence, 417 Ratio, 61 Lebesgue's Monotone Conver- Root, 60 gence, 414 Second Comparison, 59 Lerch's, 290 Theorem Littlewood's, 249 Abel's, 65, 320 Location of Zeroes, 134 Abel's (on Cauchy Product), 321 Lusin's, 444 Arzela-Ascoli, 381 Mean Value, 223 Baire Category, 169 Mean Value (for Integrals), 277 Banach's Fixed Point, 181 Mertens', 64 Banach-Steinhaus, 363 Monotone Convergence, 50, 486, Beppo Levi 's, 413 518 Bernstein Approximation, 148, Monotone Limit, 94, 101 512 Nested Intervals, 45 Bolzano-Weierstrass, 53 Newton's Binomial, 320 Bounded Convergence, 518 Open Mapping, 364 Cantor's, 32, 168 Osgood's, 184 Parseval's, 341, 460 Cantor-Bendixon, 169 Pythagorean, 371 Caratheodory's, 212 Riemann, 66 Cauchy's Condensation, 59 Riemann's Localization, 336 Cauchy-Hadamard, 313 Riemann-Darboux, 255 Closed Graph, 192, 364 Riesz Representation, 371 Complex Stone-Weierstrass, 385 Riesz-Fischer, 376 Continuous Extension, 140 Rolle's, 222 Darboux's, 222, 256 Schroder-Bernstein, 31 Dini 's, 305, 306 Second Fundamental, 280 Dirichlet's, 77 Squeeze, 91 Dominated Convergence, 489, 518 Steinhaus's, 445 Edelstein's, 206 Stone-Weierstrass, 383 Egorov's, 443, 516 Substitution, 316 Euler's, 245 Tauber's, 347 Extension, 476 Taylor's, 317 Extreme Value, 132, 191 Weierstrass Approximation, 149, F. Riesz's, 421 340 Fejer's, 339 Zermelo's Well Ordering, 11 First Fundamental, 280 Three Chords Lemma, 240 Fixed Point, 134, 143 topological property, 204 Fubini- Tonelli, 494 absolute, 187, 195 Hahn-Banach, 365 topology, 162 Heine-Borel, 116 relative, 163 Hellinger-Toeplitz, 392 torus, 194 Intermediate Value, 133 total Interval Additivity, 271 family, 362 Inverse Function, 137 mass, 468 Jordan Decomposition, 287 ordering, 9 Index 559

set, 362 total,284 Total Probability Law, 498 vector, 19 totally bounded, 189 vector addition, 19 totally disconnected, 117, 197 vector space, 19 transported distance, 178 basis of, 20 Triangle Inequality, 40, 41, 351 dimension of, 21 trichotomy, 38 vertical asymptote, 96 trigonometric function, 325 vertical slice, 193, 199 trigonometric polynomial, 329 vertical tangent, 211 trigonometric series, 330 Vieta's formula, 529 trivial extension, 399 true near, 89, 98 Wallis's Formula, 293 Washington D.C. space, 200 ultimately equal, 48 Weak Law of Large Numbers, 510 ultimately true, 48, 166 Weierstrass Approximation Theorem, ultrametric inequality, 160 149,340 ultrametric space, 160 Weierstrass M-test, 307 uncountable set , 29, 169 well ordering, 10 uniform approximation, 167 Well Ordering Axiom, 10 Uniform Boundedness Principle, 184, Well Ordering Theorem, 11 363 uniform convergence, 303 Young's inequality, 290 uniform distribution, 502 uniform limit, 303 Zorn's Lemma, 9 uniform metric, 158, 167 uniform property, 204 uniformly bounded, 70 uniformly continuous, 137, 179 uniformly differentiable, 246 uniformly equivalent metrics, 205 union, 2 unit, 4 unit element, 18, 21 unit sphere, 194 unital,20 unordered pair, 6 unordered series, 69 unordered sum, 69 associativity of, 73 upper bound, lower bound, 9 upper envelope, lower envelope, 123 upper limit , lower limit, 53, 123 Urysohn's lemma, 181

Var(X), u 2 (X ), 506 Van der Waerden, 311 variance, 506 variation bounded, 284