---
date: 2022-12-25 03:37
modification date: Sunday 25th December 2022 03:37:53
title: "Analysis"
flashcard: "Quals::Real Analysis"
---

# Analysis

- [x] What is the $M\dash$test? ✅ 2023-01-24
	- [ ] $\sum_{n\geq 0} \norm{f_n}_{\infty, A} < \infty \implies \sum f_n$ converges absolutely and uniformly on $A$.
	- [ ] Here $\norm{f_n}_{\infty, A}\da \sup_{x\in A}\abs{f_n(x)}$.
	- [ ] Equivalently, if there is a sequence $\ts{M_n}$ where $\abs{f_n(x)} \leq M_n$ for all $x\in A$ and $\sum M_n < \infty$.
- [x] Give several equivalent characterizations of completeness. ✅ 2023-01-24
	- [ ] $X$ is complete $\iff$ $X$ is Cauchy complete $\iff$ absolutely convergent implies convergent for series.
- [x] What is the Arzela-Ascoli theorem? ✅ 2023-01-24
	- [ ] A sequence $f_i$ has a uniformly convergent subsequence $\iff$ the sequence is uniformly bounded and uniformly equicontinuous.
- [x] What is Minkowski's inequality? ✅ 2023-01-24
	- [ ] For $1 \leq p < \infty$, $$\norm{f + g}_p \leq \norm{f}_p + \norm{g}_p$$
- [x] What is Young's inequality? ✅ 2023-01-24
	- [ ] For $1\leq p, q\leq r \leq \infty$ with ${1\over p} + {1\over q} - {1\over r} = 1$, then  $$\norm{f\ast g}_r \leq \norm{f}_p \norm{g}_q$$
	- [ ] Useful cases: $$\begin{aligned} \|f * g\|_1 & \leq\|f\|_1\|g\|_1 \\ \|f * g\|_p & \leq\|f\|_1\|g\|_p \\ \|f * g\|_{\infty} & \leq\|f\|_p\|g\|_q \\ \|f * g\|_{\infty} & \leq\|f\|_2\|g\|_2\end{aligned}$$
- [x] What is a Baire space? ✅ 2023-01-24
	- [ ] $X$ is a Baire space $\iff$ whenever $\theset{U_n}$ is a *countable* collection of open dense subsets of $X$, then their intersection $\intersect U_n$ is again dense.
- [x] What is a first category set? A second category? ✅ 2023-01-24
	- [ ] A subset is *first category* $\iff$ it is countable union of nowhere dense sets,  *second category* otherwise.
- [x] What does it mean for a set to be nowhere dense? ✅ 2023-01-24
	- [ ] A set is $A$ **nowhere dense** if its closure has empty interior $\qty{\bar A}^\circ$, equivalently it is not dense in *any* nonempty open set.
	- [ ] For $\RR$, every interval $I$ contains a subinterval $S\subset I$ with $S\intersect A = \emptyset$, i.e. its closure contains no intervals.
	- [ ] Intuition: elements are not tightly clustered, set is full of holes.
- [x] Give an example of a set that is not nowhere dense. ✅ 2023-01-24
	- [ ] Counterexample: $\theset{1 \over n}, \ZZ$ are nowhere dense, $\QQ, \ZZ\union \qty{(a, b)\intersect \QQ}$ is *not* nowhere dense
- [x] What is equicontinuity? Uniform equicontinuity? ✅ 2023-01-24
	- [ ] For $X, Y$ metric spaces and $\mathcal{F}$ a family of functions, $F$ is *equicontinuous at $x_0$* $\iff$ for every $\varepsilon > 0$ there exists a $\delta(\varepsilon, x_0)>0$ such that $$x\in B_\delta(x_0) \implies f_i(x) \in B_\varepsilon(f_i(x_0))$$ for all $f_i \in \mathcal{F}$.
	- [ ] The family $F$ is *uniformly equicontinuous* $\iff$ $\delta(\varepsilon)$ only depends on $\varepsilon$ and holds for any pair $x_1, x_2$ with $x_1 \in B_\delta(x_2)$.
- [x] What is the reverse triangle inequality? ✅ 2023-01-24
	- [ ] $$\abs{\, \norm{x_{}} - \norm{y} \,} \leq \norm{x-y}$$
- [x] What is a meagre set? ✅ 2023-01-24
	- [ ] A set is *meagre* $\iff$ it is a countable union of nowhere dense sets.
- [x] Characterize the set $D_f$ of discontinuities of a function. ✅ 2023-01-24
	- [ ] Always $F_\sigma$, closed, positive oscillation.
	- [ ] $f_n\to f$ with $f_n$ continuous $\implies D_f$ is meager.
	- [ ] (Lebesgue criterion) $f \in \mathcal{R}(a, b)$ and bounded $\implies D_f$ is null.
	- [ ] $f$ monotone $\implies D_f$ is countable, and additionally $f$ d$\iff$erentiable on $(a, b) \implies D_f$ is null.
- [x] What is the Baire category theorem? ✅ 2023-01-24
	- [ ] If $X$ is a complete metric space or a locally compact Hausdorff space, then $X$ is a Baire space.
	- [ ] A (non-empty) complete metric space is *not* the countable union of nowhere dense sets.
- [x] What is the Caratheodory characterization of outer measure? ✅ 2023-01-24
	- [ ] $E\subseteq \RR^n$ is measurable $\$\iff$$ for all $A\subset \RR^n$,  $$m_*(A) = m_*(E\intersect A) + m_*(E\intersect A^c)$$
- [x] What are almost disjoint sets? ✅ 2023-01-24
	- [ ] $A^\circ \intersect B^\circ = \emptyset$
- [x] What is convergence in measure? ✅ 2023-01-24
	- [ ] $$\lim _{k \rightarrow \infty} m\left(\left\{x \in E|| f_{k}(x)-f(x) |>\alpha\right\}\right)=0$$
- [x] What is a separable space? ✅ 2023-01-24
	- [ ] Contains a countable dense subset.
- [x] Is the composition of Lebesgue measurable functions again Lebesgue measurable? ✅ 2023-01-24
	- [ ] **No:** Take $f: [0, 1]\to [0, 1]$ the Cantor-Lebesgue function (monotonic and cts) and $C$ the Cantor set
	- [ ] $f(C) = [0, 1]$, so define $g(x) = f(x) +x$ so $g:[0, 1] \to [0, 2]$ (strictly monotonic and cts, so a homeomorphism), so $g\inv$ is cts and thus measurable.
	- [ ] $\mu(g(C)) = 1>0$ (because $f$ is constant on every interval in $C^c$) so $g(C) \supseteq A$ a non-measurable subset
	- [ ] $g\inv(A) \subset C$ with $\mu(C) = 0$ implies $g\inv(A)$ is a measurable set, so $\chi_{g\inv(A)}$ is a measurable function
	- [ ] Then $k\definedas \chi_{g\inv(A)} \circ g\inv$ isn't measurable since $$k\inv(1) = \qty{ (g\inv)\inv \circ \chi_{g\inv(A)} }(1) = g(g\inv(A)) = A.$$
	  - [ ] is not a measurable set.
- [x] What is a dense subset? ✅ 2023-01-24
	- [ ] A subset $A\subseteq X$ is *dense* in $X$ $\iff$ $\mathrm{cl}_X(A) = X$.
- [x] What is the uniform boundedness principle? ✅ 2023-01-24
	- [ ] If $\mathcal{F}$ is a family of bounded operators $T_n:X\to Y$ between Banach spaces with $$\forall x\in X, \qquad \sup_{T_n \in \mathcal{F}} \norm{T_n(x)}_Y < \infty$$then $\sup_{T_n\in \mathcal{F}} \norm{T_n}_X < \infty$.
	- [ ] Slogan: pointwise bounded sequences of operators are uniformly bounded.
- [x] What is the diameter of set? ✅ 2023-01-24
	- [ ] $\mathrm{diam}(A) = \sup_{x, y\in  A} \abs d(x, y)$
- [x] What is the Bolzano-Weierstrass Property ✅ 2023-01-24
	- [ ] Every sequence has a convergent subsequence
- [x] What is a complete measure? ✅ 2023-01-24
	- [ ] A measure whose domain includes all subsets of null sets.
- [x] Define $\limsup, \liminf$ for sequences of sets. What are their containments? ✅ 2023-01-24
	- [ ] $$\liminf _{n \rightarrow \infty} A_n=\bigcup_{n \geq 1} \bigcap_{j \geq n} A_j, \qquad \limsup _{n \rightarrow \infty} A_n=\bigcap_{n \geq 1} \bigcup_{j \geq n} A_j$$
	- Supremum: Union, $\limsup = \inf\sup$, in infinitely many
	- Infimum: Intersection, $\liminf = \sup\inf$, eventually in
	- $\liminf _{n \rightarrow \infty} A_n \subseteq \limsup _{n \rightarrow \infty} A_n$ since "all but finitely many" implies "infinitely often".
- [x] What is the Hahn-Banach theorem? ✅ 2023-01-24
	- [ ] If $p: V\to \RR$ is a sublinear function and $\phi: U\to \RR$ a linear functional on $U\leq V$ with $\phi \leq p$, then there exists a $p\dash$sublinear extension $\tilde \phi: V\to \RR$
- [x] What is the Riesz Representation theorem? ✅ 2023-01-24
	- [ ] For $H$ a Hilbert space and $\varphi \in H\dual$, there exists an $f\in H$ such that $x\in H \implies \varphi(x) = \inner{f}{x}$ with $\norm{f}_H = \norm{\varphi}_{H\dual}$.
- [x] What is a compact operator? ✅ 2023-01-24
	- [ ] The image of every bounded subset has compact closure.
	- [ ] Necessarily bounded
	- [ ] Closure of space of finite-rank operators in the norm topology
- [x] What is the Borel-Canteli lemma? ✅ 2023-01-24
	- [ ] $\sum P(E_n) < \infty \implies P(\limsup_n E_n) == 0$
	- [ ] If $E_n$ are independent events and $\sum P(E_n) = \infty$< then $P(\limsup_n E_n) = 1$.

## More

- [x] What does small tails mean? Absolute continuity? ✅ 2023-01-24
	- [ ] Let $f\in L^1$ and $\varepsilon> 0$.
	- [ ] **Small Tails**: there exists an $N$ such that \(\int_{B_N^c} f < \varepsilon\).
	- [ ] **Absolute Continuity**: there exists a $\delta$ such that $m(E) < \delta \implies \int_E \abs{f} < \varepsilon$.


- [ ] What is Egorov's theorem?
- [ ] How is convergence in measure related to a.e. convergence?
- [ ] What is Fatou's lemma?
- [ ] What is Egorov's theorem?
- [ ] What is a Borel set?
- [ ] What is the DCT?
- [ ] What is the MCT?
- [ ] What does it mean for a function to be Lebesgue measurable?
- [ ] What is Bernoulli's inequality?
- [ ] What is uniform continuity?
- [ ] Prove Borel-Cantelli.
- [ ] What is the Cauchy-Schwarz inequality?
- [ ] What is the Holder inequality?
- [ ] What is Bessel's inequality?
- [ ] What is Parseval's identity?
- [ ] What is the Riemann-Lebesgue lemma?
- [ ] What are $F_\sigma$ and $G_\delta$ sets?
- [ ] What is a null set?
- [ ] What does it mean for $f_n\to f$ uniformly?
- [ ] How do you show a sequence of functions converges uniformly?
- [ ] Give a function that converges pointwise but not uniformly.
- [ ] How does one negative uniform convergence?
- [ ] How are continuity and d$\iff$erentiability related?
- [ ] What is continuity of measure?
- [ ] What does it mean for a set to be measurable?
- [ ] Give a function that is Lebesgue integrable but not Riemann integrable.
- [ ] Define outer measure.
- [ ] Define a measurable function.
- [ ] What is Lusin's theorem?
- [ ] What is Chebyshev's inequality?
- [ ] Show that if $\int f = 0$ then $f=0$ a.e.
- [ ] How do you commute a sum and an integral?
- [ ] Show: Proposition: $\sum \abs{f_n} \in L^1 \implies \sum \abs{f_n(x)} < \infty$ a.e.
- [ ] Give a sequence of functions that converges uniformly, pointwise, a.e., but not in $L^1$.
- [ ] List the types of convergence in order of strength.
- [ ] What is continuity in $L^1$?
- [ ] Prove translation/dilation invariance of the Lebesgue integral.
- [ ] What is Tonelli's theorem? Fubini? Fubini-Tonelli?
- [ ] What are the four major properties of outer measure?
- [ ] Discuss integrability of $1/x^p$.
- [ ] How do you replace a sequence of sets by a sequence of disjoint sets?
- [ ] Is the pointwise limit of a sequence of bounded functions bounded?
- [ ] Give a sequence of d$\iff$erentiable functions $f_k\to f$ with $f_k' \not\to f'$.
- [ ] What are the inclusions among $L^p$ spaces? For $\ell^p$?
- [ ] What is the infinity norm?
- [ ] What is the Weierstrass approximation theorem?