Final Review for Math 421, Fall 2009

I. Technical terms you must know.

You must be able to give a careful, complete and accurate of these terms. You must also know examples of all of these terms (and their negations or combinations if that makes sense).
  1. the sequence of functions fn converges pointwise to f
  2. the sequence of functions fn converges uniformly to f
  3. a series of functions converges pointwise or uniformly
  4. (X,d) is a metric space
  5. a sequence of points in a metric space converges
  6. a sequence of points in a metric space is Cauchy
  7. an open set in a metric space
  8. a complete metric space

II. Theorems/Results/Axioms/Examples you must know.

This means you must be able to state them and to give examples of them being true and examples of how the conclusion is false if individual hypothesese are dropped (for the theorems), and discuss why (i.e., prove, albeit with a reduced standard of rigor) the claimed properties hold true (for the examples).
  1. A sequence of functions which converges pointwise but not uniformly.
  2. the book's Theorem 6.3.1 (the one about convergence of functions along with convergence of their derivatives)
  3. some famous metric spaces (Know also what the open balls look like in each of these spaces.)
  4. the construction of the real numbers R out of Dedekind cuts
  5. all of the theorems in the next section, below

III. Theorems you should be able to prove during the final:

  1. If fn converges uniformly to f and all the fn are continuous, then f is continuous also.
  2. The Weierstrass M-test
  3. the book's Theorems 6.5.1 and 6.5.2 about the sets of points where a power series converges
  4. the Axiom of Completion for the real numbers in their version as the set of Dedekind cuts

IV. Format of the final>

  1. your choice of three out of five or six of part I, above
  2. your choice of three out of five or six of part II, above
  3. your choice of two out of three or four of part III, above
  4. your choice of one or two exercises from about four or 5 (of similar style, content and difficulty to homework problems you have done in the class this term)