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Lecture Notes Kuttler October 8, 2006 Contents I Preliminary Material Set 1.1 1.2 1.3 1.4 11 11 14 17 18 Riemann Stieltjes Integral Upper And Lower Riemann Stieltjes Sums Exercises Functions Of Riemann Integrable Functions Properties Of The Integral Fundamental Theorem Of Calculus Exercises 19 19 23 24 27 31 35 Important Linear Algebra 3.1 Algebra in Fn 3.2 Subspaces Spans And Bases 3.3 An Application To Matrices 3.4 The Mathematical Theory Of Determinants 3.5 The Cayley Hamilton Theorem 3.6 An Identity Of Cauchy 3.7 Block Multiplication Of Matrices 3.8 Shur’s Theorem 3.9 The Right Polar Decomposition 3.10 The Space L (Fn , Fm ) 3.11 The Operator Norm 37 39 40 44 46 59 60 61 63 69 71 72 The 4.1 4.2 4.3 4.4 4.5 75 78 83 83 85 89 The 2.1 2.2 2.3 2.4 2.5 2.6 Theory Basic Definitions The Schroder Bernstein Theorem Equivalence Relations Partially Ordered Sets Frechet Derivative C Functions C k Functions Mixed Partial Derivatives Implicit Function Theorem More Continuous Partial Derivatives II CONTENTS Lecture Notes For Math 641 and 642 Metric Spaces And General Topological 5.1 Metric Space 5.2 Compactness In Metric Space 5.3 Some Applications Of Compactness 5.4 Ascoli Arzela Theorem 5.5 General Topological Spaces 5.6 Connected Sets 5.7 Exercises 91 Spaces Approximation Theorems 6.1 The Bernstein Polynomials 6.2 Stone Weierstrass Theorem 6.2.1 The Case Of Compact Sets 6.2.2 The Case Of Locally Compact Sets 6.2.3 The Case Of Complex Valued Functions 6.3 Exercises 93 93 95 98 100 103 109 112 115 115 117 117 120 121 122 Abstract Measure And Integration 7.1 σ Algebras 7.2 The Abstract Lebesgue Integral 7.2.1 Preliminary Observations 7.2.2 Definition Of The Lebesgue Integral For Nonnegative Measurable Functions 7.2.3 The Lebesgue Integral For Nonnegative Simple Functions 7.2.4 Simple Functions And Measurable Functions 7.2.5 The Monotone Convergence Theorem 7.2.6 Other Definitions 7.2.7 Fatou’s Lemma 7.2.8 The Righteous Algebraic Desires Of The Lebesgue Integral 7.3 The Space L1 7.4 Vitali Convergence Theorem 7.5 Exercises 125 125 133 133 The 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 157 157 163 164 169 179 179 181 185 185 189 Construction Of Measures Outer Measures Regular measures Urysohn’s lemma Positive Linear Functionals One Dimensional Lebesgue Measure The Distribution Function Completion Of Measures Product Measures 8.8.1 General Theory 8.8.2 Completion Of Product Measure Spaces 135 136 139 140 141 142 144 145 151 153 CONTENTS 8.9 Disturbing Examples 191 8.10 Exercises 193 Lebesgue Measure 9.1 Basic Properties 9.2 The Vitali Covering Theorem 9.3 The Vitali Covering Theorem (Elementary Version) 9.4 Vitali Coverings 9.5 Change Of Variables For Linear Maps 9.6 Change Of Variables For C Functions 9.7 Mappings Which Are Not One To One 9.8 Lebesgue Measure And Iterated Integrals 9.9 Spherical Coordinates In Many Dimensions 9.10 The Brouwer Fixed Point Theorem 9.11 Exercises 10 The 10.1 10.2 10.3 10.4 10.5 10.6 Lp Spaces Basic Inequalities And Properties Density Considerations Separability Continuity Of Translation Mollifiers And Density Of Smooth Functions Exercises 197 197 201 203 206 209 213 219 220 221 224 228 233 233 241 243 245 246 249 11 Banach Spaces 11.1 Theorems Based On Baire Category 11.1.1 Baire Category Theorem 11.1.2 Uniform Boundedness Theorem 11.1.3 Open Mapping Theorem 11.1.4 Closed Graph Theorem 11.2 Hahn Banach Theorem 11.3 Exercises 253 253 253 257 258 260 262 270 12 Hilbert Spaces 12.1 Basic Theory 12.2 Approximations In Hilbert Space 12.3 Orthonormal Sets 12.4 Fourier Series, An Example 12.5 Exercises 275 275 281 284 286 288 Lp 291 291 297 304 312 314 13 Representation Theorems 13.1 Radon Nikodym Theorem 13.2 Vector Measures 13.3 Representation Theorems For The Dual Space Of 13.4 The Dual Space Of C (X) 13.5 The Dual Space Of C0 (X) CONTENTS 13.6 More Attractive Formulations 316 13.7 Exercises 317 14 Integrals And Derivatives 14.1 The Fundamental Theorem Of Calculus 14.2 Absolutely Continuous Functions 14.3 Differentiation Of Measures With Respect To Lebesgue Measure 14.4 Exercises 321 321 326 331 336 15 Fourier Transforms 15.1 An Algebra Of Special Functions 15.2 Fourier Transforms Of Functions In G 15.3 Fourier Transforms Of Just About Anything 15.3.1 Fourier Transforms Of Functions In L1 (Rn ) 15.3.2 Fourier Transforms Of Functions In L2 (Rn ) 15.3.3 The Schwartz Class 15.3.4 Convolution 15.4 Exercises 343 343 344 347 351 354 359 361 363 III Complex Analysis 367 16 The Complex Numbers 369 16.1 The Extended Complex Plane 371 16.2 Exercises 372 17 Riemann Stieltjes Integrals 373 17.1 Exercises 383 18 Fundamentals Of Complex Analysis 18.1 Analytic Functions 18.1.1 Cauchy Riemann Equations 18.1.2 An Important Example 18.2 Exercises 18.3 Cauchy’s Formula For A Disk 18.4 Exercises 18.5 Zeros Of An Analytic Function 18.6 Liouville’s Theorem 18.7 The General Cauchy Integral Formula 18.7.1 The Cauchy Goursat Theorem 18.7.2 A Redundant Assumption 18.7.3 Classification Of Isolated Singularities 18.7.4 The Cauchy Integral Formula 18.7.5 An Example Of A Cycle 18.8 Exercises 385 385 387 389 390 391 398 401 403 404 404 407 408 411 418 422 CONTENTS 19 The Open Mapping Theorem 19.1 A Local Representation 19.1.1 Branches Of The Logarithm 19.2 Maximum Modulus Theorem 19.3 Extensions Of Maximum Modulus Theorem 19.3.1 Phragmen Lindelăof Theorem 19.3.2 Hadamard Three Circles Theorem 19.3.3 Schwarz’s Lemma 19.3.4 One To One Analytic Maps On The Unit Ball 19.4 Exercises 19.5 Counting Zeros 19.6 An Application To Linear Algebra 19.7 Exercises 425 425 427 429 431 431 433 434 435 436 438 442 446 20 Residues 20.1 Rouche’s Theorem And The Argument Principle 20.1.1 Argument Principle 20.1.2 Rouche’s Theorem 20.1.3 A Different Formulation 20.2 Singularities And The Laurent Series 20.2.1 What Is An Annulus? 20.2.2 The Laurent Series 20.2.3 Contour Integrals And Evaluation Of Integrals 20.3 The Spectral Radius Of A Bounded Linear Transformation 20.4 Exercises 449 452 452 455 456 457 457 460 464 473 475 21 Complex Mappings 21.1 Conformal Maps 21.2 Fractional Linear Transformations 21.2.1 Circles And Lines 21.2.2 Three Points To Three Points 21.3 Riemann Mapping Theorem 21.3.1 Montel’s Theorem 21.3.2 Regions With Square Root Property 21.4 Analytic Continuation 21.4.1 Regular And Singular Points 21.4.2 Continuation Along A Curve 21.5 The Picard Theorems 21.5.1 Two Competing Lemmas 21.5.2 The Little Picard Theorem 21.5.3 Schottky’s Theorem 21.5.4 A Brief Review 21.5.5 Montel’s Theorem 21.5.6 The Great Big Picard Theorem 21.6 Exercises 479 479 480 480 482 483 484 486 490 490 492 493 495 498 499 503 505 506 508 CONTENTS 22 Approximation By Rational Functions 22.1 Runge’s Theorem 22.1.1 Approximation With Rational Functions 22.1.2 Moving The Poles And Keeping The Approximation 22.1.3 Merten’s Theorem 22.1.4 Runge’s Theorem 22.2 The Mittag-Leffler Theorem 22.2.1 A Proof From Runge’s Theorem 22.2.2 A Direct Proof Without Runge’s Theorem 22.2.3 Functions Meromorphic On C 22.2.4 A Great And Glorious Theorem About Simply Connected Regions 22.3 Exercises 511 511 511 513 513 518 520 520 522 524 23 Infinite Products 23.1 Analytic Function With Prescribed Zeros 23.2 Factoring A Given Analytic Function 23.2.1 Factoring Some Special Analytic Functions 23.3 The Existence Of An Analytic Function With Given Values 23.4 Jensen’s Formula 23.5 Blaschke Products 23.5.1 The Mă untz-Szasz Theorem Again 23.6 Exercises 529 533 538 540 542 546 549 552 554 24 Elliptic Functions 24.1 Periodic Functions 24.1.1 The Unimodular Transformations 24.1.2 The Search For An Elliptic Function 24.1.3 The Differential Equation Satisfied By 24.1.4 A Modular Function 24.1.5 A Formula For λ 24.1.6 Mapping Properties Of λ 24.1.7 A Short Review And Summary 24.2 The Picard Theorem Again 24.3 Exercises 563 564 568 571 574 576 582 584 592 596 597 ℘ 524 528 A The Hausdorff Maximal Theorem 599 A.1 Exercises 603 Copyright c 2005, Part I Preliminary Material 24.3 EXERCISES 597 Thus h is an entire function which misses the two values and If h is not constant, then by Lemma 24.30 there exists a function, g analytic on C which has values in the upper half plane, P+ such that λ ◦ g = h However, g must be a constant because there exists ψ an analytic map on the upper half plane which maps the upper half plane to B (0, 1) You can use the Riemann mapping theorem or more simply, ψ (z) = z−i z+i Thus ψ ◦ g equals a constant by Liouville’s theorem Hence g is a constant and so h must also be a constant because λ (g (z)) = h (z) This proves f is a constant also This proves the theorem 24.3 Exercises Show the set of modular transformations is a group Also show those modular transformations which are congruent mod to the identity as described above is a subgroup Suppose f is an elliptic function with period module M If {w1 , w2 } and {w1 , w2 } are two bases, show that the resulting period parallelograms resulting from the two bases have the same area Given a module of periods with basis {w1 , w2 } and letting a typical element of this module be denoted by w as described above, consider the product z σ (z) ≡ z 1− e(z/w)+ (z/w) w w=0 Show this product converges uniformly on compact sets, is an entire function, and satisfies σ (z) /σ (z) = ζ (z) where ζ (z) was defined above as a primitive of ℘ (z) and is given by ζ (z) = + z w=0 z + + z − w w2 w Show ζ (z + wi ) = ζ (z) + η i where η i is a constant Let Pa be the parallelogram shown in the following picture ✘ ✘ ✘✘✘ ✂ ✘ ✘ ✘ ✘ w2 ✘✘ ✘ ✂ ✂ ✘ s ✘✘✘ ✘ ✘ ✂ ✂ ✘✂✘ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✘ ✘ ✂ ✂s w1 ✂ ✂ ✘✘✘✂ ✂ ✘ ✂ 0✘✘✘ ✘ ✂s ✘✘✘ ✂ ✘ ✘ ✘ ✂ ✘✘ s✘ a 598 ELLIPTIC FUNCTIONS ζ (z) dz = where the contour is taken once around the Show that 2πi ∂Pa parallelogram in the counter clockwise direction Next evaluate this contour integral directly to obtain Legendre’s relation, η w2 − η w1 = 2πi For σ defined in Problem 3, explain the following steps For j = 1, σ (z + wj ) σ (z) = ζ (z + wj ) = ζ (z) + η j = + ηj σ (z + wj ) σ (z) Therefore, there exists a constant, Cj such that σ (z + wj ) = Cj σ (z) eηj z Next show σ is an odd function, (σ (−z) = −σ (z)) and then let z = −wj /2 η j wj to find Cj = −e and so σ (z + wj ) = −σ (z) eηj (z+ wj ) (24.41) Show any even elliptic function, f with periods w1 and w2 for which is neither a pole nor a zero can be expressed in the form n f (0) k=1 ℘ (z) − ℘ (ak ) ℘ (z) − ℘ (bk ) where C is some constant Here ℘ is the Weierstrass function which comes from the two periods, w1 and w2 Hint: You might consider the above function in terms of the poles and zeros on a period parallelogram and recall that an entire function which is elliptic is a constant Suppose f is any elliptic function with {w1 , w2 } a basis for the module of periods Using Theorem 24.9 and 24.41 show that there exists constants a1 , · · ·, an and b1 , · · ·, bn such that for some constant C, n f (z) = C k=1 σ (z − ak ) σ (z − bk ) Hint: You might try something like this: By Theorem 24.9, it follows that if {αk } are the zeros and {bk } the poles in an appropriate period parallelogram, αk − bk equals a period Replace αk with ak such that ak − bk = Then use 24.41 to show that the given formula for f is bi periodic Anyway, you try to arrange things such that the given formula has the same poles as f Remember an entire elliptic function equals a constant Show that the map τ → − τ1 maps l2 onto the curve, C in the above picture on the mapping properties of λ 10 Modify the proof of Theorem 24.23 to show that λ (Ω)∩{z ∈ C : Im (z) < 0} = ∅ The Hausdorff Maximal Theorem The purpose of this appendix is to prove the equivalence between the axiom of choice, the Hausdorff maximal theorem, and the well-ordering principle The Hausdorff maximal theorem and the well-ordering principle are very useful but a little hard to believe; so, it may be surprising that they are equivalent to the axiom of choice First it is shown that the axiom of choice implies the Hausdorff maximal theorem, a remarkable theorem about partially ordered sets A nonempty set is partially ordered if there exists a partial order, ≺, satisfying x≺x and if x ≺ y and y ≺ z then x ≺ z An example of a partially ordered set is the set of all subsets of a given set and ≺≡⊆ Note that two elements in a partially ordered sets may not be related In other words, just because x, y are in the partially ordered set, it does not follow that either x ≺ y or y ≺ x A subset of a partially ordered set, C, is called a chain if x, y ∈ C implies that either x ≺ y or y ≺ x If either x ≺ y or y ≺ x then x and y are described as being comparable A chain is also called a totally ordered set C is a maximal chain if whenever C is a chain containing C, it follows the two chains are equal In other words C is a maximal chain if there is no strictly larger chain Lemma A.1 Let F be a nonempty partially ordered set with partial order ≺ Then assuming the axiom of choice, there exists a maximal chain in F Proof: Let X be the set of all chains from F For C ∈ X , let SC = {x ∈ F such that C∪{x} is a chain strictly larger than C} If SC = ∅ for any C, then C is maximal Thus, assume SC = ∅ for all C ∈ X Let f (C) ∈ SC (This is where the axiom of choice is being used.) Let g(C) = C ∪ {f (C)} 599 600 THE HAUSDORFF MAXIMAL THEOREM Thus g(C) C and g(C) \ C ={f (C)} = {a single element of F} A subset T of X is called a tower if ∅∈T, C ∈ T implies g(C) ∈ T , and if S ⊆ T is totally ordered with respect to set inclusion, then ∪S ∈ T Here S is a chain with respect to set inclusion whose elements are chains Note that X is a tower Let T0 be the intersection of all towers Thus, T0 is a tower, the smallest tower Are any two sets in T0 comparable in the sense of set inclusion so that T0 is actually a chain? Let C0 be a set of T0 which is comparable to every set of T0 Such sets exist, ∅ being an example Let B ≡ {D ∈ T0 : D C0 and f (C0 ) ∈ / D} The picture represents sets of B As illustrated in the picture, D is a set of B when D is larger than C0 but fails to be comparable to g (C0 ) Thus there would be more than one chain ascending from C0 if B = ∅, rather like a tree growing upward in more than one direction from a fork in the trunk It will be shown this can’t take place for any such C0 by showing B = ∅ ·f (C0 ) C0 D This will be accomplished by showing T0 ≡ T0 \ B is a tower Since T0 is the smallest tower, this will require that T0 = T0 and so B = ∅ Claim: T0 is a tower and so B = ∅ Proof of the claim: It is clear that ∅ ∈ T0 because for ∅ to be contained in B it would be required to properly contain C0 which is not possible Suppose D ∈ T0 The plan is to verify g (D) ∈ T0 Case 1: f (D) ∈ C0 If D ⊆ C0 , then since both D and {f (D)} are contained in C0 , it follows g (D) ⊆ C0 and so g (D) ∈ / B On the other hand, if D C0 , then since D ∈ T0 , f (C0 ) ∈ D and so g (D) also contains f (C0 ) implying g (D) ∈ / B These are the only two cases to consider because C0 is comparable to every set of T0 Case 2: f (D) ∈ / C0 If D C0 it can’t be the case that f (D) ∈ / C0 because if this were so, g (D ) would not compare to C0 D ·f (C0 ) C0 ·f (D) Hence if f (D) ∈ / C0 , then D ⊇ C0 If D = C , then f (D) = f (C0 ) ∈ g (D) so 601 g (D) ∈ / B Therefore, assume D C0 Then, since D is in T0 , f (C0 ) ∈ D and so f (C0 ) ∈ g (D) Therefore, g (D) ∈ T0 Now suppose S is a totally ordered subset of T0 with respect to set inclusion Then if every element of S is contained in C0 , so is ∪S and so ∪S ∈ T0 If, on the other hand, some chain from S, C, contains C0 properly, then since C ∈ / B, f (C0 ) ∈ C ⊆ ∪S showing that ∪S ∈ / B also This has proved T0 is a tower and since T0 is the smallest tower, it follows T0 = T0 This has shown roughly that no splitting into more than one ascending chain can occur at any C0 which is comparable to every set of T0 Next it is shown that every element of T0 has the property that it is comparable to all other elements of T0 This is done by showing that these elements which possess this property form a tower Define T1 to be the set of all elements of T0 which are comparable to every element of T0 (Recall the elements of T0 are chains from the original partial order.) Claim: T1 is a tower Proof of the claim: It is clear that ∅ ∈ T1 because ∅ is a subset of every set Suppose C0 ∈ T1 It is necessary to verify that g (C0 ) ∈ T1 Let D ∈ T0 (Thus D ⊆ C0 or else D C0 )and consider g (C0 ) ≡ C0 ∪ {f (C0 )} If D ⊆ C0 , then D ⊆ g (C0 ) so g (C0 ) is comparable to D If D C0 , then D ⊇ g (C0 ) by what was just shown (B = ∅) Hence g (C0 ) is comparable to D Since D was arbitrary, it follows g (C0 ) is comparable to every set of T0 Now suppose S is a chain of elements of T1 and let D be an element of T0 If every element in the chain, S is contained in D, then ∪S is also contained in D On the other hand, if some set, C, from S contains D properly, then ∪S also contains D Thus ∪S ∈ T since it is comparable to every D ∈ T0 This shows T1 is a tower and proves therefore, that T0 = T1 Thus every set of T0 compares with every other set of T0 showing T0 is a chain in addition to being a tower Now ∪T0 , g (∪T0 ) ∈ T0 Hence, because g (∪T0 ) is an element of T0 , and T0 is a chain of these, it follows g (∪T0 ) ⊆ ∪T0 Thus ∪T0 ⊇ g (∪T0 ) ∪T0 , a contradiction Hence there must exist a maximal chain after all This proves the lemma If X is a nonempty set,≤ is an order on X if x ≤ x, and if x, y ∈ X, then either x ≤ y or y ≤ x and if x ≤ y and y ≤ z then x ≤ z ≤ is a well order and say that (X, ≤) is a well-ordered set if every nonempty subset of X has a smallest element More precisely, if S = ∅ and S ⊆ X then there exists an x ∈ S such that x ≤ y for all y ∈ S A familiar example of a well-ordered set is the natural numbers 602 THE HAUSDORFF MAXIMAL THEOREM Lemma A.2 The Hausdorff maximal principle implies every nonempty set can be well-ordered Proof: Let X be a nonempty set and let a ∈ X Then {a} is a well-ordered subset of X Let F = {S ⊆ X : there exists a well order for S} Thus F = ∅ For S1 , S2 ∈ F , define S1 ≺ S2 if S1 ⊆ S2 and there exists a well order for S2 , ≤2 such that (S2 , ≤2 ) is well-ordered and if y ∈ S2 \ S1 then x ≤2 y for all x ∈ S1 , and if ≤1 is the well order of S1 then the two orders are consistent on S1 Then observe that ≺ is a partial order on F By the Hausdorff maximal principle, let C be a maximal chain in F and let X∞ ≡ ∪C Define an order, ≤, on X∞ as follows If x, y are elements of X∞ , pick S ∈ C such that x, y are both in S Then if ≤S is the order on S, let x ≤ y if and only if x ≤S y This definition is well defined because of the definition of the order, ≺ Now let U be any nonempty subset of X∞ Then S ∩ U = ∅ for some S ∈ C Because of the definition of ≤, if y ∈ S2 \ S1 , Si ∈ C, then x ≤ y for all x ∈ S1 Thus, if y ∈ X∞ \ S then x ≤ y for all x ∈ S and so the smallest element of S ∩ U exists and is the smallest element in U Therefore X∞ is well-ordered Now suppose there exists z ∈ X \ X∞ Define the following order, ≤1 , on X∞ ∪ {z} x ≤1 y if and only if x ≤ y whenever x, y ∈ X∞ x ≤1 z whenever x ∈ X∞ Then let C = {S ∈ C or X∞ ∪ {z}} Then C is a strictly larger chain than C contradicting maximality of C Thus X \ X∞ = ∅ and this shows X is well-ordered by ≤ This proves the lemma With these two lemmas the main result follows Theorem A.3 The following are equivalent The axiom of choice The Hausdorff maximal principle The well-ordering principle A.1 EXERCISES 603 Proof: It only remains to prove that the well-ordering principle implies the axiom of choice Let I be a nonempty set and let Xi be a nonempty set for each i ∈ I Let X = ∪{Xi : i ∈ I} and well order X Let f (i) be the smallest element of Xi Then f∈ Xi i∈I A.1 Exercises Zorn’s lemma states that in a nonempty partially ordered set, if every chain has an upper bound, there exists a maximal element, x in the partially ordered set x is maximal, means that if x ≺ y, it follows y = x Show Zorn’s lemma is equivalent to the Hausdorff maximal theorem Let X be a vector space Y ⊆ X is a Hamel basis if every element of X can be written in a unique way as a finite linear combination of elements in Y Show that every vector space has a Hamel basis and that if Y, Y1 are two Hamel bases of X, then there exists a one to one and onto map from Y to Y1 ↑ Using the Baire category theorem of the chapter on Banach spaces show that any Hamel basis of a Banach space is either finite or uncountable ↑ Consider the vector space of all polynomials defined on [0, 1] Does there exist a norm, ||·|| defined on these polynomials such that with this norm, the vector space of polynomials becomes a Banach space (complete normed vector space)? Index C functions, 78 Cc∞ , 246 Ccm , 246 Fσ sets, 126 Gδ , 255 Gδ sets, 126 L1loc , 321 Lp compactness, 250 π systems, 185 σ algebra, 125 Borel measure, 163 Borel sets, 125 bounded continuous linear functions, 255 bounded variation, 373 branch of the logarithm, 428 Brouwer fixed point theorem, 224, 279 Browder’s lemma, 289 Cantor diagonalization procedure, 103 Cantor function, 229 Cantor set, 228 Caratheodory, 157 Caratheodory’s procedure, 158 Cartesian coordinates, 38 Casorati Weierstrass theorem, 408 Cauchy general Cauchy integral formula, 414 integral formula for disk, 393 Cauchy Riemann equations, 387 Cauchy Schwarz inequality, 275 Cauchy sequence, 72 Cayley Hamilton theorem, 59 chain rule, 76 change of variables general case, 220 characteristic function, 131 characteristic polynomial, 59 closed graph theorem, 261 closed set, 105 closure of a set, 106 cofactor, 53 compact, 95 compact set, 107 complete measure space, 158 completion of measure space, 181 Abel’s theorem, 398 absolutely continuous, 326 adjugate, 55 algebra, 117 analytic continuation, 492, 594 Analytic functions, 385 approximate identity, 246 at most countable, 16 automorphic function, 580 axiom of choice, 11, 15, 229 axiom of extension, 11 axiom of specification, 11 axiom of unions, 11 Banach space, 237 Banach Steinhaus theorem, 257 basis of module of periods, 568 Bessel’s inequality, 286, 289 Big Picard theorem, 507 Blaschke products, 549 Bloch’s lemma, 495 block matrix, 61 Borel Cantelli lemma, 155 Borel measurable, 229 604 INDEX conformal maps, 391, 480 connected, 109 connected components, 110 continuous function, 106 convergence in measure, 155 convex set, 276 convex functions, 250 convolution, 247, 361 Coordinates, 37 countable, 16 counting zeros, 438 Cramer’s rule, 56 cycle, 414 Darboux, 34 Darboux integral, 34 derivatives, 76 determinant, 48 product, 52 transpose, 50 differential equations Peano existence theorem, 123 dilations, 480 Dini derivates, 338 distribution function, 179 dominated convergence theorem, 149 doubly periodic, 566 dual space, 266 duality maps, 273 Egoroff theorem, 131 eigenvalues, 59, 443, 446 elementary factors, 533 elliptic, 566 entire, 403 epsilon net, 95, 100 equality of mixed partial derivatives, 85 equivalence class, 17 equivalence relation, 17 essential singularity, 409 Euler’s theorem, 561 exchange theorem, 41 605 exponential growth, 364 extended complex plane, 371 Fatou’s lemma, 143 finite intersection property, 99, 108 finite measure space, 126 Fourier series uniform convergence, 272 Fourier transform L1 , 352 fractional linear transformations, 480, 485 mapping three points, 482 Frechet derivative, 75 Fresnel integrals, 471 Fubini’s theorem, 189 function, 14 function element, 492, 594 functional equations, 584 fundamental theorem of algebra, 404 fundamental theorem of calculus, 33, 323, 325 Gamma function, 251 gamma function, 555 gauge function, 263 Gauss’s formula, 556 Gerschgorin’s theorem, 442 Gram determinant, 283 Gram matrix, 283 Gramm Schmidt process, 64 great Picard theorem, 506 Hadamard three circles theorem, 433 Hahn Banach theorem, 264 Hardy Littlewood maximal function, 321 Hardy’s inequality, 251 harmonic functions, 390 Hausdorff maximal principle, 18, 201, 263 Hausdorff maximal theorem, 599 Hausdorff metric, 114 Hausdorff space, 104 Heine Borel theorem, 98, 113 Hermitian, 67 Hilbert space, 275 606 Holder’s inequality, 233 homotopic to a point, 525 implicit function theorem, 85, 88, 89 indicator function, 131 infinite products, 529 inner product space, 275 inner regular measure, 163 inverse function theorem, 89, 90 inverses and determinants, 54 inversions, 480 isogonal, 390, 479 isolated singularity, 408 James map, 268 Jensen’s formula, 546 Jensens inequality, 251 Laplace expansion, 53 Laplace transform, 230, 364 Laurent series, 460 Lebesgue set, 325 Lebesgue decomposition, 291 Lebesgue measure, 197 Lebesgue point, 323 limit point, 105 linear combination, 40, 51 linearly dependent, 40 linearly independent, 40 Liouville theorem, 403 little Picard theorem, 596 locally compact , 107 Lusin’s theorem, 250 matrix left inverse, 55 lower triangular, 56 non defective, 67 normal, 67 right inverse, 55 upper triangular, 56 maximal function measurability, 337 maximal function strong estimates, 337 maximum modulus theorem, 429 INDEX mean value theorem for integrals, 35 measurable, 157 Borel, 128 measurable function, 128 pointwise limits, 128 measurable functions Borel, 154 combinations, 131 measurable sets, 126, 158 measure space, 126 Mellin transformations, 468 meromorphic, 410 Merten’s theorem, 513 Minkowski functional, 272 Minkowski’s inequality, 239 minor, 53 Mittag Leffler, 472, 540 mixed partial derivatives, 83 modular function, 578, 580 modular group, 509, 568 module of periods, 564 mollifier, 246 monotone convergence theorem, 140 monotone functions differentiable, 339 Montel’s theorem, 483, 505 multi-index, 83, 343 Neumann series, 473 nonmeasurable set, 229 normal family of functions, 485 normal topological space, 105 nowhere differentiable functions, 270 one point compactification, 107, 166 open cover, 107 open mapping theorem, 258, 425 open sets, 104 operator norm, 72, 255 order, 555 order of a pole, 409 order of a zero, 401 order of an elliptic function, 566 orthonormal set, 284 INDEX outer measure, 154, 157 outer regular measure, 163 parallelogram identity, 288 partial derivative, 79 partial order, 18, 262 partially ordered set, 599 partition, 19 partition of unity, 168 period parallelogram, 566 Phragmen Lindelof theorem, 431 pi systems, 185 Plancherel theorem, 356 point of density, 336 polar decomposition, 303 pole, 409 polynomial, 343 positive and negative parts of a measure, 333 positive linear functional, 169 power series analytic functions, 397 power set, 11 precompact, 107, 122 primitive, 381 principal branch of logarithm, 429 principal ideal, 544 product topology, 106 projection in Hilbert space, 278 properties of integral properties, 31 Radon Nikodym derivative, 294 Radon Nikodym Theorem σ finite measures, 294 finite measures, 291 rank of a matrix, 56 real Schur form, 65 reflexive Banach Space, 269 reflexive Banach space, 311 region, 401 regular family of sets, 337 regular measure, 163 regular topological space, 105 removable singularity, 408 607 residue, 449 resolvent set, 473 Riemann criterion, 23 Riemann integrable, 22 continuous, 113 Riemann integral, 22 Riemann sphere, 371 Riemann Stieltjes integral, 22 Riesz map, 281 Riesz representation theorem C0 (X), 315 Hilbert space, 280 locally compact Hausdorff space, 169 Riesz Representation theorem C (X), 314 Riesz representation theorem Lp finite measures, 304 Riesz representation theorem Lp σ finite case, 310 Riesz representation theorem for L1 finite measures, 308 right polar decomposition, 69 Rouche’s theorem, 455 Runge’s theorem, 518 Sard’s lemma, 217 scalars, 39 Schottky’s theorem, 503 Schroder Bernstein theorem, 15 Schwarz formula, 399 Schwarz reflection principle, 423 Schwarz’s lemma, 486 self adjoint, 67 separated, 109 separation theorem, 273 sets, 11 Shannon sampling theorem, 366 simple function, 136 Sm´ıtal, 338 Sobolev Space embedding theorem, 365 equivalent norms, 365 Sobolev spaces, 365 span, 40, 51 608 spectral radius, 474 stereographic projection, 372, 504 Stirling’s formula, 557 strict convexity, 274 subspace, 40 support of a function, 167 Tietze extention theorem, 124 topological space, 104 total variation, 297, 326 totally bounded set, 95 totally ordered set, 599 translation invariant, 199 translations, 480 trivial, 40 uniform boundedness theorem, 257 uniform convergence, 370 uniform convexity, 274 uniformly bounded, 100, 505 uniformly equicontinuous, 100, 505 uniformly integrable, 151 unimodular transformations, 568 upper and lower sums, 20 Urysohn’s lemma, 164 variational inequality, 278 vector measures, 297 Vitali convergence theorem, 152, 251 Vitali covering theorem, 202, 205, 206, 208 Vitali coverings, 206, 208 Vitali theorem, 509 weak convergence, 274 Weierstrass approximation theorem, 117 Stone Weierstrass theorem, 118 Weierstrass M test, 370 Weierstrass P function, 573 well ordered sets, 601 winding number, 411 Young’s inequality, 233, 319 zeta function, 557 INDEX Bibliography [1] Adams R Sobolev Spaces, Academic Press, New York, San Francisco, London, 1975 [2] Alfors, Lars Complex Analysis, McGraw Hill 1966 [3] Apostol, T M., Mathematical Analysis, Addison Wesley Publishing Co., 1969 [4] Apostol, T M., Calculus second edition, Wiley, 1967 [5] Apostol, T M., Mathematical Analysis, Addison Wesley Publishing Co., 1974 [6] Ash, Robert, Complex Variables, Academic Press, 1971 [7] Baker, Roger, Linear Algebra, Rinton Press 2001 [8] Bergh J and Lă ofstră om J Interpolation Spaces, Springer Verlag 1976 [9] Bledsoe W.W , Am Math Monthly vol 77, PP 180-182 1970 [10] Bruckner A , Bruckner J., and Thomson B., Real Analysis Prentice Hall 1997 [11] Conway J B Functions of one Complex variable Second edition, Springer 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