Tags: #qualifying_exam #active_projects #qual_real_analysis

Study Guide

References:

• Folland’s “Real Analysis: Modern Techniques”, Ch.1
• Stein and Shakarchi Ch.1, Ch.2

Convergence Tips/Tricks

• Our favorite tools: metrics and norms!
  • So show things are equal by showing \({\left\lvert {x-y} \right\rvert} = 0\). Know the triangle inequality by heart!
• Uniform convergence:
  • Negating: find a bad \({\varepsilon}\) and a single bad point \(x\).
  • Showing a sum converges uniformly: remember that \(\sum_{k\geq 1} a_k\) is defined to be \(\lim_{N\to\infty} \sum_{k\leq N} a_k\). So the trick is to define \(f_n(x) := \sum_{k\leq n} a_k\) and then apply the usual criteria above.
  • It’s sometimes useful to trade the \(\forall x\) in the definition with \(\sup_{x\in X} {\left\lvert {f_n(x) - f(x)} \right\rvert} < {\varepsilon}\) instead.
• Compare and contrast to pointwise convergence, which is strictly weaker:
  • The main difference: pointwise can depend on the \(x\) and the \({\varepsilon}\), but uniform needs one \({\varepsilon}\) that works for all \(x\) simultaneously.
  • Note uniform implies pointwise but not conversely.
• The sup norm: \({\left\lVert {f} \right\rVert}_\infty := \sup_{x\in X} {\left\lvert {f_n(x)} \right\rvert}\)
  • A useful way to force uniform convergence: bound your sequence uniformly by a sequence that goes to zero:
• Sups and infs: sup is the least upper bound, inf is the greatest lower bound.
• The \(p-\)test: \begin{align*} \sum_{n\geq 1} {1 \over n^p} < \infty \iff p>1 \end{align*}
• Useful fact: convergent sums have small tails, i.e.  \begin{align*} \sum_{n\geq 1} a_n < \infty \implies \lim_{N\to\infty}\sum_{n\geq N} a_n = 0 \end{align*}
• So try bounding things from above by the tail of a sum!
• If you can’t bound by a tail: as long as you have control over the coefficients, you can pick them to make the sum to converge “fast enough”.
  • Example: for a fixed \({\varepsilon}\), choose \(a_n = 1/2^n\). Note that \(\sum_{n\geq 1} 1/2^n = 1\), so choose \(a_n := {\varepsilon}/2^n\): \begin{align*} \cdots \leq \sum_{n\geq 1} a_n := \sum_{n\geq 1} {{\varepsilon}\over 2^n} = {\varepsilon}\to 0 \end{align*}
• The \({\varepsilon}/3\) trick:
• The \(M{\hbox{-}}\)test:

Measure Theory

• \(F_\sigma\) sets: unions of closed sets (\(F\) for fermi, French for closed. Sigma for sums, ie unions)
• \(G_\delta\) sets: intersections of open sets
• \(\sigma\) algebras: closed under complements, countable intersections, countable unions
• Some of the most useful properties of measures:

• The proof of continuity of measure contains a very useful trick: replace a sequence of sets \(\left\{{E_k}\right\}\) with a sequence of disjoint sets that either union or intersect to the same thing.
  • Example: if \(A_1 \subseteq A_2 \subseteq \cdots\), set \(F_1=A_1\) and \(F_k = A_k \setminus A_{k-1}\) for \(k\geq 2\). Then \(\bigcup_{k\geq 1} A_k = \coprod_{k\geq 1} F_k\).
• Occasionally you need some properties of outer measures:

• Outer measure for \({\mathbf{R}}^n\): you consider all collections of cubes that cover your set, sum up their volumes, and take the infimum over all such collections:

• “Almost everywhere blah” : the set where blah does not happen has measure zero.
• “Infinitely many/all but finitely many” types of sets, which show up in Borel-Cantelli style problems

• Lemmas that sometimes show up on quals:

Fubini-Tonelli

Quick statement:

Explained in Stein and Shakarchi (Fubini, which requires integrability)

And Tonelli, which only requires measurability:

A more precise statement from Folland:

Some things that qual questions are commonly based on:

Qual Problems

Suggested by Peter Woolfitt!

Spring 2012

Fall 2016.2

Fall 2018.5

Spring 2019.4: This is an expanded version of Fall 2018 #5 above.