Course Notes
Reaching for Infinity
a first course in analysis
Steve Trettel
A first course in real analysis, told as the story of infinite processes: why we need them, how they fail, and the ideas — completeness, convergence, uniformity — that make them safe to use.
Contents
Preface
Notation
Numbers
1 Measurement
2 Completeness
3 Limits
4 Convergence
5 General Sequences
6 Arithmetic of the Infinite
Functions
7 Inputs and Outputs
8 Continuity
9 Differentiation
10 Integration
11 The Calculus
12 Elementary Functions
13 π
Spaces
14 Function Space
15 Extending the Integral
16 Completion
17 Infinite Dimensional Geometry
18 Infinite Dimensional Coordinates
19 Beyond Functions
Solutions
20 Differential Equations
21 Variational Problems
22 Eigenfunction Expansion
23 Fourier Transform
24 Fundamental Solutions