Basic Analysis Kenneth Kuttler April 16, 2001 2 Contents 1 Basic Set theory 9 1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2 The Schroder Bernstein theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Linear Algebra 15 2.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Inner product spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.6 The characteristic polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.7 The rank of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3 General topology 43 3.1 Compactness in metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2 Connected sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3 The Tychonoff theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4 Spaces of Continuous Functions 61 4.1 Compactness in spaces of continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.2 Stone Weierstrass theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5 Abstract measure and Integration 71 5.1 σ Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.2 Monotone classes and algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.4 The Abstract Lebesgue Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.5 The space L 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.6 Double sums of nonnegative terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.7 Vitali convergence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.8 The ergodic theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3 4 CONTENTS 6 The Construction Of Measures 97 6.1 Outer measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.2 Positive linear functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 7 Lebesgue Measure 113 7.1 Lebesgue measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7.2 Iterated integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7.3 Change of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 7.4 Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.5 The Lebesgue integral and the Riemann integral . . . . . . . . . . . . . . . . . . . . . . . . . 126 7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 8 Product Measure 131 8.1 Measures on infinite products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 8.2 A strong ergodic theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 8.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 9 Fourier Series 147 9.1 Definition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 9.2 Pointwise convergence of Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 9.2.1 Dini’s criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 9.2.2 Jordan’s criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 9.2.3 The Fourier cosine series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 9.3 The Cesaro means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 9.4 Gibb’s phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 9.5 The mean square convergence of Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . 160 9.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 10 The Frechet derivative 169 10.1 Norms for finite dimensional vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 10.2 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 10.3 Higher order derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 10.4 Implicit function theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 10.5 Taylor’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 10.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 11 Change of variables for C 1 maps 199 11.1 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 11.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 12 The L p Spaces 209 12.1 Basic inequalities and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 12.2 Density of simple functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 12.3 Continuity of translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 12.4 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 12.5 Mollifiers and density of smooth functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 12.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 CONTENTS 5 13 Fourier Transforms 223 13.1 The Schwartz class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 13.2 Fourier transforms of functions in L 2 (R n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 13.3 Tempered distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 13.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 14 Banach Spaces 243 14.1 Baire category theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 14.2 Uniform boundedness closed graph and open mapping theorems . . . . . . . . . . . . . . . . . 246 14.3 Hahn Banach theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 14.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 15 Hilbert Spaces 257 15.1 Basic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 15.2 Orthonormal sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 15.3 The Fredholm alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 15.4 Sturm Liouville problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 15.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 16 Brouwer Degree 277 16.1 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 16.2 Definitions and elementary properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 16.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 16.4 The Product formula and Jordan separation theorem . . . . . . . . . . . . . . . . . . . . . . . 289 16.5 Integration and the degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 17 Differential forms 297 17.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 17.2 The integration of differential forms on manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 298 17.3 Some examples of orientable manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 17.4 Stokes theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 17.5 A generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 17.6 Surface measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 17.7 Divergence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 17.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 18 Representation Theorems 319 18.1 Radon Nikodym Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 18.2 Vector measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 18.3 Representation theorems for the dual space of L p . . . . . . . . . . . . . . . . . . . . . . . . . 325 18.4 Riesz Representation theorem for non σ finite measure spaces . . . . . . . . . . . . . . . . . . 330 18.5 The dual space of C (X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 18.6 Weak ∗ convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 18.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 19 Weak Derivatives 345 19.1 Test functions and weak derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 19.2 Weak derivatives in L p loc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 19.3 Morrey’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 19.4 Rademacher’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 19.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 6 CONTENTS 20 Fundamental Theorem of Calculus 359 20.1 The Vitali covering theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 20.2 Differentiation with respect to Lebesgue measure . . . . . . . . . . . . . . . . . . . . . . . . . 361 20.3 The change of variables formula for Lipschitz maps . . . . . . . . . . . . . . . . . . . . . . . . 364 20.4 Mappings that are not one to one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 20.5 Differential forms on Lipschitz manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 20.6 Some examples of orientable Lipschitz manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 376 20.7 Stoke’s theorem on Lipschitz manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 20.8 Surface measures on Lipschitz manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 20.9 The divergence theorem for Lipschitz manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 383 20.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 21 The complex numbers 393 21.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 21.2 The extended complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 21.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 22 Riemann Stieltjes integrals 399 22.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 23 Analytic functions 409 23.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 23.2 Examples of analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 23.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 24 Cauchy’s formula for a disk 415 24.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 25 The general Cauchy integral formula 423 25.1 The Cauchy Goursat theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 25.2 The Cauchy integral formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 25.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 26 The open mapping theorem 433 26.1 Zeros of an analytic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 26.2 The open mapping theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 26.3 Applications of the open mapping theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 26.4 Counting zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 26.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 27 Singularities 443 27.1 The Laurent series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 27.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 28 Residues and evaluation of integrals 451 28.1 The argument principle and Rouche’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 460 28.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 28.3 The Poisson formulas and the Hilbert transform . . . . . . . . . . . . . . . . . . . . . . . . . 462 28.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 28.5 Infinite products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 28.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 CONTENTS 7 29 The Riemann mapping theorem 477 29.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 30 Approximation of analytic functions 483 30.1 Runge’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 30.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 A The Hausdorff Maximal theorem 489 A.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 8 CONTENTS Basic Set theory We think of a set as a collection of things called elements of the set. For example, we may consider the set of integers, the collection of signed whole numbers such as 1,2,-4, etc. This set which we will believe in is denoted by Z. Other sets could be the set of people in a family or the set of donuts in a display case at the store. Sometimes we use parentheses, { } to specify a set. When we do this, we list the things which are in the set between the parentheses. For example the set of integers between -1 and 2, including these numbers could be denoted as {−1, 0, 1, 2}. We say x is an element of a set S, and write x ∈ S if x is one of the things in S. Thus, 1 ∈ {−1, 0, 1, 2, 3}. Here are some axioms about sets. Axioms are statements we will agree to believe. 1. Two sets are equal if and only if they have the same elements. 2. To every set, A, and to every condition S (x) there corresponds a set, B, whose elements are exactly those elements x of A for which S (x) holds. 3. For every collection of sets there exists a set that contains all the elements that belong to at least one set of the given collection. 4. The Cartesian product of a nonempty family of nonempty sets is nonempty. 5. If A is a set there exists a set, P (A) such that P (A) is the set of all subsets of A. These axioms are referred to as the axiom of extension, axiom of specification, axiom of unions, axiom of choice, and axiom of powers respectively. It seems fairly clear we should want to believe in the axiom of extension. It is merely saying, for example, that {1, 2, 3} = {2, 3, 1} since these two sets have the same elements in them. Similarly, it would seem we would want to specify a new set from a given set using some “condition” which can be used as a test to determine whether the element in question is in the set. For example, we could consider the set of all integers which are multiples of 2. This set could be specified as follows. {x ∈ Z : x = 2y for some y ∈ Z}. In this notation, the colon is read as “such that” and in this case the condition is being a multiple of 2. Of course, there could be questions about what constitutes a “condition”. Just because something is grammatically correct does not mean it makes any sense. For example consider the following nonsense. S = {x ∈ set of dogs : it is colder in the mountains than in the winter}. We will leave these sorts of considerations however and assume our conditions make sense. The axiom of unions states that if we have any collection of sets, there is a set consisting of all the elements in each of the sets in the collection. Of course this is also open to further consideration. What is a collection? Maybe it would be better to say “set of sets” or, given a set whose elements are sets there exists a set whose elements 9 10 BASIC SET THEORY consist of exactly those things which are elements of at least one of these sets. If S is such a set whose elements are sets, we write the union of all these sets in the following way. ∪{A : A ∈ S} or sometimes as ∪S. Something is in the Cartesian product of a set or “family” of sets if it consists of a single thing taken from each set in the family. Thus (1, 2, 3) ∈ {1, 4, .2}×{1, 2, 7}×{4, 3, 7, 9} because it consists of exactly one element from each of the sets which are separated by ×. Also, this is the notation for the Cartesian product of finitely many sets. If S is a set whose elements are sets, we could write  A∈S A for the Cartesian product. We can think of the Cartesian product as the set of choice functions, a choice function being a function which selects exactly one element of each set of S. You may think the axiom of choice, stating that the Cartesian product of a nonempty family of nonempty sets is nonempty, is innocuous but there was a time when many mathematicians were ready to throw it out because it implies things which are very hard to believe. We say A is a subset of B and write A ⊆ B if every element of A is also an element of B. This can also be written as B ⊇ A. We say A is a proper subset of B and write A ⊂ B or B ⊃ A if A is a subset of B but A is not equal to B, A = B. The intersection of two sets is a set denoted as A ∩ B and it means the set of elements of A which are also elements of B. The axiom of specification shows this is a set. The empty set is the set which has no elements in it, denoted as ∅. The union of two sets is denoted as A ∪B and it means the set of all elements which are in either of the sets. We know this is a set by the axiom of unions. The complement of a set, (the set of things which are not in the given set ) must be taken with respect to a given set called the universal set which is a set which contains the one whose complement is being taken. Thus, if we want to take the complement of a set A, we can say its complement, denoted as A C ( or more precisely as X \A) is a set by using the axiom of specification to write A C ≡ {x ∈ X : x /∈ A} The symbol /∈ is read as “is not an element of”. Note the axiom of specification takes place relative to a given set which we believe exists. Without this universal set we cannot use the axiom of specification to speak of the complement. Words such as “all” or “there exists” are called quantifiers and they must be understood relative to some given set. Thus we can speak of the set of all integers larger than 3. Or we can say there exists an integer larger than 7. Such statements have to do with a given set, in this case the integers. Failure to have a reference set when quantifiers are used turns out to be illogical even though such usage may be grammatically correct. Quantifiers are used often enough that there are symbols for them. The symbol ∀ is read as “for all” or “for every” and the symbol ∃ is read as “there exists”. Thus ∀∀∃∃ could mean for every upside down A there exists a backwards E. 1.1 Exercises 1. There is no set of all sets. This was not always known and was pointed out by Bertrand Russell. Here is what he observed. Suppose there were. Then we could use the axiom of specification to consider the set of all sets which are not elements of themselves. Denoting this set by S, determine whether S is an element of itself. Either it is or it isn’t. Show there is a contradiction either way. This is known as Russell’s paradox. [...]... the Schroder Bernstein theorem Theorem 1.2 Let f : X → Y and g : Y → X be two mappings Then there exist sets A, B, C, D, such that A ∪ B = X, C ∪ D = Y, A ∩ B = ∅, C ∩ D = ∅, f (A) = C, g (D) = B 12 BASIC SET THEORY The following picture illustrates the conclusion of this theorem X Y f A B = g(D) ' E C = f (A) g D Proof: We will say A0 ⊆ X satisfies P if whenever y ∈ Y \ f (A0 ) , g (y) ∈ A0 Note... exists α : N →X which is one to one and onto Let β : X ×Y → N be defined by β ((x, y)) ≡ α−1 (x) Then by Corollary 1.5, there is a one to one and onto mapping from X × Y to N This proves the corollary 14 BASIC SET THEORY Theorem 1.9 If X and Y are at most countable, then X ∪ Y is at most countable Proof: Let X = {xi }∞ , Y = {yj }∞ and consider the following array consisting of X ∪ Y and path i=1 j=1 through... validity of this identity is equivalent to the existence of an inner product which determines the norm as described above These sorts of considerations are topics for more advanced courses on functional analysis Definition 2.19 We say a basis for an inner product space, {u1 , · · ·, un } is an orthonormal basis if (uk , uj ) = δ kj ≡ 1 if k = j 0 if k = j 24 LINEAR ALGEBRA Note that if a list of vectors... proves the proposition in the case when there are no repeated numbers in the ordered list, (r1 , · · ·, rn ) However, if there is a repeat, say the rth row equals the sth row, then the reasoning of (2.16) -( 2.17) shows that A (r1 , · · ·, rn ) = 0 and we also know that sgn (r1 , · · ·, rn ) = 0 so the formula holds in this case also 34 LINEAR ALGEBRA Corollary 2.35 We have the following formula for det . Basic Analysis Kenneth Kuttler April 16, 2001 2 Contents 1 Basic Set theory 9 1.1 Exercises . . . . . . . . . . . . . . . . c n such that u 2 = c 1 u 1 + n  k= 2 c k v k . By the assumption that {u 1 , · · ·, u m } is linearly independent, we know that at least one of the c k for k ≥ 2 is non zero. Without loss of. begin with {u 1 , · · ·, u k }. If span (u 1 , · · ·, u k ) = V, we are done. If not, there exists a vector, u k+ 1 /∈ span (u 1 , · · ·, u k ) . Then {u 1 , ···, u k , u k+ 1 } is also linearly independent.