Math 414 Analysis II

Key Concepts

You should be comfortable with all of the key definitions.
You should be able to state and apply all of the key results.
You should be able to sketch the proof of the key results marked with (*).
You should also be familiar with the key concepts from Math 413

Chapter 7: Integration

Key Definitions

upper and lower Riemann sums
Riemann integrable functions (this is sometimes called the Riemann-Darboux integral)
General Riemann sums
step functions
Measure zero sets

Key Results

characterization of integrable functions
continuous functions are integrable (*)
properties of integrals
approximation of integrable functions by continuous functions (*)
equivalence of Riemann-Darboux and Riemann integrals
Fundamental Theorem of Calculus (two versions) (*)
Lebesgue criteria for integrability

Chapter 2: Infinite Series

Key Definitions

Convergent series
Absolutely and conditionally convergent series
Rearrangements

Key Results

Cauchy Criteria (*)
Comparison Test(*)
Ratio Test (*)
Integral Test (*)
Alternating Series Test (*)
Rearragments of conditionally and absolutely convregent series
Multiplication of absolutely convergenet series

Chapter 6: Sequences of Functions

Key Definitions

Pointwise convergent sequences
Uniformly convergent sequences
Series of functions
Power series
Radius of convergence
Taylor series

Key Results

The limit of a uniformly convergent sequence of continuous functions is continuous.(*)
The limit of a uniformly convergent sequence of integrable functions is integrable.(*)
The existence of nowhere differentiable continuous functions.
Criteria for differentiablity of limits of sequences of differentiable functions.
Weierstrass M-test(*)
Abel's Theorem
Continuity (*), differentiability and integrability(*) results for power series.
Remainder Theorem

Contraction Mapping Theorem

Key Definitions

Contraction Mappings

Key Results

Contraction Mapping Theorem for R (*)
Contraction Mapping Theorem for Spaces of Continuous Functions
Existence and Uniqueness Theorem for Ordinary Differential Equations
Implicit Function Theorem
Inverse Function Theorem

Baire Category in Function Spaces

Key Definitions

Open and Closed sets in C([0,1])
Dense, nowhere dense and meager sets in C([0,1])
Piecewise linear functions

Key Results

Density of Piecewise Linear Functions in C([0,1])(*)
Baire Category Theorem for C([0,1])
The set of functions in C([0,1]) that are differentiable somewhere is meager.