Section F.2 List of definitions and results
1.1 Metrics, Norms and Inner Products
Lemma 1.4 Cauchy–Schwarz inequality
Example 1.5 Euclidean spaces
Example 1.6 Discrete metric
Example 1.7 Supremum norm
Lemma 1.8 Reverse triangle inequality
Example 1.13 \(C^0([a,b])\)
1.2 Open and Closed Sets
Corollary 1.27 Open sets are unions of balls
Lemma 1.28 Open sets in metric subspaces
1.3 Convergence and Completeness
Example 1.33 Uniform convergence
1.4 Continuity
Theorem 1.45 Continuity of compositions
Theorem 1.46 Sequential continuity
Proposition 1.47 Continuity of basic operations
1.5 The Space \(C_\bdd(X)\)
Theorem 1.51 Uniform limit theorem
1.6 Functions and higher-order functions
Example 1.55 Functions vs. formulas
Example 1.59 Some simple higher-order functions
Example 1.61 Continuity of a higher-order function
1.7 Isometries
1.8 Cardinality
Definition 1.64 Cardinality
Definition 1.65 Finite sets
Definition 1.66 Countable and uncountable
Corollary 1.68 Countable sets and sequences
Corollary 1.69 Countability and subsets
Lemma 1.70 Countability and products
Lemma 1.72 Countability and unions
2.1 Dense Sets
Lemma 2.4 Continuous extension
2.2 Existence of the Completion
2.3 Uniqueness of the Completion
2.4 Normed and Inner Product Spaces
Theorem 2.12 Completions of normed and inner product spaces
3.1 \(C^k([a,b])\)
3.2 Hölder Spaces
3.3 Integration Review
Theorem 3.7 Properties of the Riemann integral
Corollary 3.8 Estimating integrals
Lemma 3.9 Inertia principle for integrals
3.4 \(L^2([a,b])\)
4.1 Sequential Compactness
Lemma 4.3 Sequential compactness and metric subspaces
4.2 Total Boundedness
Lemma 4.8 Total boundedness and metric subspaces
4.3 Compactness
Lemma 4.19 Compactness and metric subspaces
4.4 Relative Compactness
4.5 The Arzelà–Ascoli Theorem
Theorem 4.29 Arzelà–Ascoli
5.1 The Weierstrass Approximation Theorem
Theorem 5.1 Weierstrass approximation theorem
Theorem 5.5 Approximation by trigonometric polynomials
5.2 The Stone–Weierstrass Theorem
Theorem 5.14 Stone–Weierstrass
6.1 The Theorem
Theorem 6.1 Baire category theorem
