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!