Analysis

Sets & Functions

• Sets:
  • Set Theory Fundamentals (operations, algebra, results)
    • De Morgan's Laws
  • Set Functions (images, pre-images; pre-images preserve unions and complements, images do not preserve complements)
  • limsup and liminf of sets (see also Limsup, Liminf and Measure)
  • [[convex-concave-sets]], Convex Set
• Relations and cardinality:
  • Relations (equivalence relations, orderings)
  • Set Countability (cardinality, countability, uncountability of \(\mathbb{R}\))
  • Axiom of Choice (used in Vitalli Non-Measurability Theorem)

Real Analysis

• The real line:
  • Real Numbers (axioms, completeness, order)
  • Supremum & Infimum
  • Extended reals \(\overline{\mathbb{R}}=[-\infty,\infty]\) (topology, arithmetic conventions)
• Sequences:
  • Sequence Convergence (limits, Cauchy sequences, subsequences)
  • limsup and liminf of sequences (inequalities, sums, negatives)
• Functions on the reals:
  • Real Functions (limits, continuity, differentiability, integrability)
    • Continuity of Real Functions
    • Differentiability
  • Semicontinuity: lower and upper semicontinuous functions, semicontinuous envelopes
  • Convergence of Functions (pointwise, uniform)
  • Images and Preimages of Functions
  • Convex Function
• Riemann integration:
  • Riemann Integral (upper and lower sums, Lebesgue criterion)
  • Functions which are not Riemann integrable (the Dirichlet function)

Metric Spaces and Topology

• Metric spaces:
  • Metric Spaces (metrics, examples such as \(d_\infty\) and \(d_1\) on \(C([0,1])\))
  • Cauchy Sequence, Completeness (\(C([0,1])\) is complete under \(d_\infty\) but not under \(d_1\))
  • Epsilon Ball
  • Lipschitz Continuity
• Topology:
  • Topology (open sets, closed sets, neighborhoods, interior, closure)
  • Point Set Topologies (bases, Hausdorff spaces, continuity, homeomorphisms)
  • Boundary Points
  • Structure of open sets in \(\mathbb{R}\) (countable unions of open intervals)
• Compactness:
  • Compactness (compact sets are closed in Hausdorff spaces, continuous bijections from compact spaces are homeomorphisms)
  • Cantor set (closed, nowhere dense, uncountable, measure zero)

Measure Theory

• Measurable spaces:
  • Measure Spaces
    • Algebra, \(\sigma\)-algebra, monotonicity, finite and countable additivity, set generated \(\sigma\)-algebra.
    • Measures, properties of measures (continuity from above and below, null sets, completeness).
  • Generated Sigma Algebra
  • Borel σ-Algebra and Measure
    • Borel sets, Borel Sigma algebra
    • Borel measure
    • Generators of the Borel \(\sigma\)-algebra on \(\mathbb{R}\) (open, closed, half-open intervals and rays)
  • Product σ-Algebra
  • Pi System, Monotone Class Theorem, Dynkin's π-λ Theorem
• Constructing measures:
  • Outer Measures (measurable sets, extension of a measure to an outer measure)
  • Caratheodory Extension Theorem
    • Extending from pre-measure to full measure.
  • Lebesgue-Stieltjes measures and locally finite Borel measures on \(\mathbb{R}\) (regularity)
• Lebesgue measure:
  • Exploration - Lebesgue Measure
  • Vitalli Non-Measurability Theorem
    • Impossible to construct measure without \(\sigma\)-algebra
  • Cantor set and Cantor function
  • Lebesgue versus Borel measurable sets (completeness)
• Measurable functions:
  • Measurable Function
  • Borel and Lebesgue measurable functions, extended-valued functions
  • Simple functions and approximation by simple functions
  • Almost everywhere equivalence

Integration

• The Lebesgue integral:
  • Lebesgue Integral (simple functions, nonnegative functions, \(L^1\))
  • Relationship with the Riemann integral (Riemann Integral)
• Convergence theorems:
  • Monotone Convergence Theorem
  • Fatou Lemma
  • Dominated Convergence Theorem
  • Uniform Integrability (later)
• Modes of convergence:
  • Pointwise, uniform and almost everywhere (Convergence of Functions, Convergence Almost Surely)
  • Convergence in measure (metrizable on finite measure spaces) (Convergence in Probability)
  • Egoroff’s theorem and Lusin’s theorem
• Inequalities:
  • Markov Inequality, Chebyshev Inequality
  • Jensen Inequality
  • Hölder Inequality, Minkowski Inequality, Cauchy-Schwartz Inequality
  • Triangle Inequality
• Product measures:
  • Product measures, Tonelli’s and Fubini’s theorems
  • Counterexamples: completeness of product spaces, non-\(\sigma\)-finite spaces
• Differentiation of measures (later):
  • Radon-Nikodym Derivative
  • Signed measures

Functional Analysis

• Normed and inner product spaces (later):
  • Vector Spaces, Linear Maps
  • Normed Space (Banach spaces)
  • Inner Product Space, Hilbert Spaces
• \(L^p\) spaces:
  • Lp Norm, Lp Spaces
  • Hölder Inequality, Minkowski Inequality
  • Completeness of \(L^p\) (Riesz-Fischer)
• Overview: Functional Analysis

Backlinks

• Lipschitz Continuity
• Real Functions
• Topics