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 \(E\subseteq {\mathbb{R}}\). 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 \({\mathbb{R}}^d\)?
Let \(X\) be a subset of \({\mathbb{R}}^d\). 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 \({\mathbb{R}}\) are opened, closed, both, or neither:
- \({\mathbb{Q}}\)
- \({\mathbb{Z}}\)
- \(\left\{{1}\right\}\)
- \(\left\{{p \in {\mathbb{Z}}^{\geq 0} {~\mathrel{\Big\vert}~}p\text{ is prime}}\right\}\)
- \(\left\{{ {1\over n} {~\mathrel{\Big\vert}~}n\in {\mathbb{Z}}^{\geq 0}}\right\}\)
- \(\left\{{ {1\over n} {~\mathrel{\Big\vert}~}n\in {\mathbb{Z}}^{\geq 0}}\right\} \cup\left\{{0}\right\}\)

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 \begin{align*}a_n \, {\overset{{n} \to\infty}\longrightarrow 0} \, \end{align*} . - Show that convergent sequences have small tails in the following sense: \begin{align*}\sum_{n > N} a_n \, {\overset{{N} \to\infty}\longrightarrow 0} \, \end{align*} . - Is this a necessary and sufficient condition for convergence? - State the ratio, root, integral, and alternating series tests. - Prove that the harmonic series diverges - Derive a formula for the sum of a geometric series. - State and prove the \(p{\hbox{-}}\)test. - What does it mean for a series to converge absolutely? - Find a sequence that converges but not absolutely.

Continuity and Discontinuity

What does it mean for a function to be uniformly continuous on a set?
Is it possible for a function \(f:{\mathbb{R}}\to {\mathbb{R}}\) to be discontinuous precisely on the rationals \({\mathbb{Q}}\)? If so, produce such a function, if not, why?
- Can the set of discontinuities be precisely the irrationals \({\mathbb{R}}\setminus{\mathbb{Q}}\)?
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.