Tags: #qualifying_exam #active_projects #qual_real_analysis

Week 1: Preliminaries

Topics

• Continuity and uniform continuity
  • Pathological functions and sequences of functions
• Convergence
  • The Cauchy criterion
  • Uniform convergence
  • The $M$-Test
• Heine-Borel
• Normed spaces
• Series and sequences,
  • Convergence
  • Small tails,
  • limsup and liminf,
  • Cauchy criteria for sums and integrals
• Variation and bounded variation

Things that don’t explicitly appear in qual problems

• Baire category theorem,
• Nowhere density,
• Basic inequalities (triangle, Cauchy-Schwarz)
• Concepts from Calculus
  • Mean value theorem
  • Taylor expansion
  • Taylor’s remainder theorem
  • Intermediate value theorem
  • Extreme value theorem
  • Rolle’s theorem
  • Riemann integrability
• Weierstrass approximation

Background / Warmup / Review

• Derive the reverse triangle inequality from the triangle inequality.
• Let $E\subseteq {\mathbf{R}}$. Define $\sup E$ and $\inf E$.
• What is the Archimedean property?

Metric Spaces / Topology

• What does it mean for a metric space to be complete?
• Give two or more equivalently definitions for compactness in a complete metric space.
• What is an interior point? An isolated point? A limit point?
• What does it mean for a set to be open? Closed?
• What is the closure of a subspace $E\subseteq X$?
• What does it mean for $E\subseteq X$ to be a dense subspace?
• What does it mean for a family of sets to form a basis for a topology?
  • What is a basis for the standard topology on ${\mathbf{R}}^d$?
• Let $X$ be a subset of ${\mathbf{R}}^d$. Prove the Heine-Borel theorem:
  • Show that $X$ compact $\implies X$ is closed
  • Show that $X$ compact $\implies X$ is bounded
  • Show that a closed subset of a compact set must be bounded.
  • Show that if $X$ closed and bounded $\implies X$ is compact.
• Find an example of a metric space with a closed and bounded subspace that is not compact.
  • How can this be modified to obtain a necessary and sufficient condition?
• Determine if the following subsets of ${\mathbf{R}}$ are opened, closed, both, or neither:
  • ${\mathbf{Q}}$
  • ${\mathbf{Z}}$
  • $\left\{{1}\right\}$
  • $\left\{{p \in {\mathbf{Z}}^{\geq 0} {~\mathrel{\Big\vert}~}p\text{ is prime}}\right\}$
  • $\left\{{ {1\over n} {~\mathrel{\Big\vert}~}n\in {\mathbf{Z}}^{\geq 0}}\right\}$
  • $\left\{{ {1\over n} {~\mathrel{\Big\vert}~}n\in {\mathbf{Z}}^{\geq 0}}\right\} \cup\left\{{0}\right\}$

Sequences

• Can a convergent sequence of real numbers have a subsequence converging to a different limit?
• What does it mean for a sequence of functions to converge pointwise and to converge uniformly?
  • Give an example of a sequence that converges pointwise but not uniformly.
• Prove that every sequence admits a monotone subsequence.
• Prove the monotone convergence theorem for sequences.
• Prove the Bolzano-Weierstrass Theorem.

Series

– What does it mean for a series to converge? How can you check this?

• What does it mean for a series to converge uniformly? What do you have to show to prove it does not converge uniformly?
• Show that if $\sum_{n\in {\mathbb{N}}} a_n < \infty$ converges, then `

Continuity and Discontinuity

• What does it mean for a function to be uniformly continuous on a set?

• Is it possible for a function $f:{\mathbf{R}}\to {\mathbf{R}}$ to be discontinuous precisely on the rationals ${\mathbf{Q}}$? If so, produce such a function, if not, why?

  • Can the set of discontinuities be precisely the irrationals ${\mathbf{R}}\setminus{\mathbf{Q}}$?

• Find a sequence of continuous functions that does not converge uniformly, but still has a pointwise limit that is continuous.

Exercises

• Find a function that is differentiable but not continuously differentiable.

• Prove the uniform limit theorem: a uniform limit of continuous function is continuous.

• Show that the uniform limit of bounded functions is uniformly bounded.

• Construct sequences of functions $\left\{{f_n}\right\}_{n\in {\mathbb{N}}}$ and $\left\{{g_n}\right\}_{n\in {\mathbb{N}}}$ which converge uniformly on some set $E$, and yet their product sequence $\left\{{h_n}\right\}_{n\in {\mathbb{N}}}$ with $h_n \coloneqq f_n g_n$ does not converge uniformly.

  • Show that if $f_n, g_n$ are additionally bounded, then $h_n$ does converge uniformly.

• Find a sequence of functions such that $$\begin{align*}\frac{d}{d x} \lim _{n \rightarrow \infty} f_{n}(x) \neq \lim _{n \rightarrow \infty} \frac{d}{d x} f_{n}(x)\end{align*}$$

• Find a uniform limit of differentiable functions that is not differentiable.

• Prove that the Cantor set is a Borel set.

• Show the Cantor ternary set is totally disconnected; that is show it contains no nonempty open interval.

Qual Questions

• • Note: outline omitted!