Section F.1 List of exercises
1 Fundamental Concepts
Exercise 1.1.1 Iterated triangle inequality
Exercise 1.1.2 Spaces with three points
Exercise 1.1.3 Are these metric spaces?
Exercise 1.1.4 The inner product is not the metric
Exercise 1.1.5 (PS2) Some norms on \(\R^2\)
Exercise 1.1.6 (PS2) Supremum norm
Exercise 1.1.7 Calculating supremum norms
Exercise 1.1.8 (PS2) Estimating supremum norms
Exercise 1.1.9 (PS2) Product spaces
Exercise 1.1.10 Normed spaces are unbounded
Exercise 1.1.11 Bounded sets in normed spaces
Exercise 1.2.1 (PS3) Inclusions of balls
Exercise 1.2.2 (PS3) Open sets in metric subspaces
Exercise 1.2.3 Closed vs. closure
Exercise 1.2.4 Interior, closure, and set operations
Exercise 1.2.5 Open and closed sets in \(C^0([a,b])\)
Exercise 1.3.1 (PS3) Some basic limits
Exercise 1.3.2 (PS3) Convergence in product spaces
Exercise 1.3.3 (PS4) Cauchy implies bounded
Exercise 1.3.4 (PS4) Pairs of Cauchy sequences
Exercise 1.3.5 Cauchy sequences and double limits
Exercise 1.3.6 Limit points
Exercise 1.4.1 (PS4) Metric is Lipschitz
Exercise 1.4.2 (PS4) Image of closure under continuous map
Exercise 1.4.3 Continuity in terms of open sets
Exercise 1.5.1 Uniform limit of continuous maps
Exercise 1.6.1 (PS4) Compositions of higher-order functions
Exercise 1.7.1 An isometry between \(\R^2\) with two different norms
Exercise 1.7.2 Evaluation map on \(B(S)\)
2 Completion of Metric Spaces
Exercise 2.1.1 (PS5) Dense sets in product spaces
Exercise 2.1.2 (PS5) Uniform continuity and Cauchy sequences
Exercise 2.1.3 (PS5) Continuous extensions
Exercise 2.1.4 (PS6) \(\C\) is separable
Exercise 2.1.5 Open subsets of separable spaces
Exercise 2.1.6 Separable spaces are second countable
Exercise 2.2.1 (PS6) Equivalence relation in the proof of Theorem 2.10
Exercise 2.2.2 (PS5) Triangle inequality in Theorem 2.10
Exercise 2.2.3 (PS5) Completion vs. closure
Exercise 2.4.1 Vector addition and Cauchy sequences
Exercise 2.4.2 (PS6) Scalar multiplication and Cauchy sequences
3 Some Function Spaces
Exercise 3.1.1 (PS6) Calculating and estimating \(C^k\) norms
Exercise 3.1.2 Uniform continuity of derivatives
Exercise 3.2.1 Hölder continuity in metric spaces
Exercise 3.2.2 (PS7) Regularity of \(x \mapsto x^\alpha\)
Exercise 3.2.4 \(C^{0,\alpha}\) is non-separable
Exercise 3.2.5 (PS8) \(C^{1,\alpha}([a,b])\)
Exercise 3.2.6 (PS7) Inclusions between Hölder spaces
Exercise 3.3.1 (PS6) Integrals involving piecewise-linear functions
Exercise 3.4.2 (PS7) Calculating and estimating \(L^2\) norms
Exercise 3.4.3 (PS8) \(L^2\) convergence is not pointwise convergence
Exercise 3.4.4 \(L^2\) on different intervals
4 Compact Sets
Exercise 4.1.1 (PS8) Compactness and minimisation
Exercise 4.1.2 Bounding functions on compact spaces
Exercise 4.1.3 Weierstrass extreme value theorem
Exercise 4.1.4 Nested compact sets
Exercise 4.2.1 Total boundedness for subsets
Exercise 4.3.1 (PS9) Direct proof of non-compactness
Exercise 4.3.2 Compactness and metric subspaces
Exercise 4.3.3 (PS9) Continuity on compact spaces
Exercise 4.3.4 (PS9) Images of compact sets
Exercise 4.3.5 (PS8) A closed and bounded set which is not compact
Exercise 4.4.1 (PS9) Relative compactness and subsequences
Exercise 4.5.1 (PS9) Finite sets are equicontinuous
Exercise 4.5.3 (PS10) A set which is not equicontinuous
Exercise 4.5.4 (PS10) Compactness and indefinite integrals
5 Uniform Approximation
Exercise 5.1.1 (PS10) \(C^0([a,b])\) is separable
Exercise 5.1.2 Trigonometric polynomials dense in \(L^2([a,b])\)
Exercise 5.2.1 (PS11) Some non-examples
Exercise 5.2.2 Lattice operations are continuous
Exercise 5.2.3 Piecewise linear functions
Exercise 5.2.4 Trigonometric polynomials and complex numbers
Exercise 5.2.5 (PS11) Separating points
Exercise 5.2.6 A convergence criterion for series
Exercise 5.2.7 Another convergence criterion for series
Exercise 5.2.8 Proof of Proposition 5.15
6 The Baire Category Theorem
Exercise 6.1.1 (PS11) Simple examples
Exercise 6.1.2 Nowhere dense sets and set operations
Exercise 6.1.3 Interiors, closures, and complements
Exercise 6.1.4 (PS11) Baire for linear subspaces
Exercise 6.1.5 A fact about \(\Q\)
Exercise B.1 Quantifiers and limits
Exercise B.2 Suprema and limits
Exercise B.3 Inequalities and limits
Exercise H.5.1 Question 1
Exercise H.5.2 Question 2
Exercise H.7.1 Question 1
Exercise H.7.2 Question 2
Exercise H.7.3 Question 3
Exercise H.7.4 Question 4
