Spaces, Functions and Change

Contents

1. 1
   Metric Spaces

   Distance between points, strings, functions and places on Earth, and the topology it defines.

   38 min read

2. 2
   Completeness and Contraction

   Complete spaces and the contraction mapping principle, the engine behind every existence theorem to come.

   34 min read

3. 3
   Compactness

   Compact sets, the contradiction–compactness method, and the three ways compactness is lost.

   32 min read

4. 4
   Connectedness

   Connected and path-connected sets, and the intermediate value theorem in general.

   24 min read

5. 5
   Uniform Convergence and Arzelà–Ascoli

   Spaces of functions, Weierstrass approximation, and the compactness theorem behind every later compactness theorem.

   38 min read

6. 6
   Power Series, Exponentials and Bump Functions

   Analytic functions, exp and log, the matrix exponential, and smooth functions that are not analytic.

   35 min read

7. 7
   Fourier Series and the First Heat Equation

   Fourier’s solution of heat flow on a ring: smoothing, energy decay and the first heat kernel.

   37 min read

8. 8
   Calculus in Several Variables

   The derivative as a linear map, done rigorously, and the Hessian at a maximum.

   37 min read

9. 9
   The Inverse and Implicit Function Theorems

   When a nonlinear equation can be solved: linearise, invert, and use a contraction.

   34 min read

10. 10
    Ordinary Differential Equations

    Existence, uniqueness, Gronwall, comparison, blow-up and invariant sets.

    39 min read

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