1: Tools for Analysis
- Page ID
- 49098
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)This chapter discusses various mathematical concepts and constructions which are central to the study of the many fundamental results in analysis. Generalities are kept to a minimum in order to move quickly to the heart of analysis: the structure of the real number system and the notion of limit. The reader should consult the bibliographical references for more details.
- 1.1: Basic Concepts of Set Theory
- This page covers foundational concepts in set theory, defining sets as collections of elements and introducing key operations like union, intersection, and difference. It explains notation for set membership and subsets, including the empty set, and presents examples of familiar sets. Additionally, the page discusses the Cartesian product of sets, illustrating the formation of ordered pairs and emphasizing their distinction from sets.
- 1.2: Functions
- This page offers a comprehensive overview of functions in mathematics, detailing definitions, properties, and types such as injective, surjective, and bijective. It explains foundational concepts like domain, codomain, and range, while also introducing important theorems regarding inverse images, composite functions, and set relationships. Exercises reinforce understanding of these properties and their proofs, particularly regarding finite and infinite sets and their corresponding relationships.
- 1.3: The Natural Numbers and Mathematical Induction
- This page explores the Well-Ordering Property of natural numbers and introduces mathematical induction, detailing how to prove propositions through a base case and an inductive step. It includes examples for concepts such as the sum of natural numbers and divisibility. Additionally, it covers strong induction, emphasizing its conditions and applications through examples like sums of squares and the Fibonacci sequence, along with exercises for further practice in induction techniques.
- 1.4: Ordered Field Axioms
- This page covers the axiomatic foundations of real numbers, detailing ordered field axioms, completeness, and properties of addition and multiplication, including identity and inverses. It contrasts real, rational, and integer numbers, deriving properties and presenting proofs. Additionally, it explores the absolute value concept, defining irrational numbers and highlighting key properties like non-negativity and the triangle inequality with geometric interpretations.
- 1.5: The Completeness Axiom for the Real Numbers
- This page covers upper and lower bounds in real numbers, detailing the definitions of least upper bounds (suprema) and greatest lower bounds (infima). It explains the completeness axiom for real numbers and distinguishes them from other ordered fields. The text includes examples, discusses the extended real number system, and outlines operations involving infinity. It emphasizes that every nonempty subset has a supremum and infimum, while providing exercises to apply these concepts.
- 1.6: Applications of the Completeness Axiom
- This page covers the fundamental properties of real numbers as defined by the Completeness Axiom, including the Archimedean Property and the notion of unboundedness. It establishes the density of rational and irrational numbers, proving the irrationality of \(\sqrt{2}\). Exercises further reinforce these concepts by exploring rational approximations and the interactions between rational and irrational numbers.