# 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**
**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.