Tags: #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
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Our favorite tools: metrics and norms!
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- So show things are equal by showing \({\left\lvert {x-y} \right\rvert} = 0\). Know the triangle inequality by heart!
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Uniform convergence:
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- 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.
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Compare and contrast to pointwise convergence, which is strictly weaker:
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- 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.
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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:
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- 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!
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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*}
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The \({\varepsilon}/3\) trick:
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The \(M{\hbox{-}}\)test:
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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:
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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 \(\displaystyle\bigcup_{k\geq 1} A_k = \displaystyle\coprod_{k\geq 1} F_k\).
- Occasionally you need some properties of outer measures:
- Outer measure for \({\mathbb{R}}^n\): 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.
