Convergence and Continuity

\begin{align*} \limsup_n a_n = \lim_{n\to \infty} \sup_{j\geq n} a_j &= \inf_{n\geq 0} \sup_{j\geq n} a_j \\ \liminf_n a_n = \lim_{n\to \infty} \inf_{j\geq n} a_j &= \sup_{n\geq 0} \inf_{j\geq n} a_j .\end{align*}

A function \(f: {\mathbb{R}}\to {\mathbb{R}}\) is continuous on \(X\) iff for all \(x_0\in X\), \begin{align*} &\forall \varepsilon \quad \exists \delta(\varepsilon, x_0) \text{ such that }\quad \forall y, {\left\lvert {x_0 - y} \right\rvert} < \delta &&\implies {\left\lvert {f(x_0) - f(y)} \right\rvert} < \varepsilon \\ \iff &\forall \varepsilon \quad \exists \delta(\varepsilon, x_0) \text{ such that }\quad \forall h, {\left\lvert {h} \right\rvert} < \delta &&\implies {\left\lvert {f(x_0) - f(x_0 \pm h)} \right\rvert} < \varepsilon .\end{align*}

\(f\) is uniformly continuous on \(X\) iff

\begin{align*} &\forall \varepsilon \quad \exists \delta(\varepsilon) \text{ such that }\quad \forall x, y, \in X \quad {\left\lvert {x - y} \right\rvert} < \delta &&\implies {\left\lvert {f(x) - f(y)} \right\rvert} < \varepsilon \\ \iff &\forall \varepsilon \quad \exists \delta(\varepsilon) \text{ such that} \quad \, \forall x, h, \quad {\left\lvert {h} \right\rvert} < \delta &&\implies {\left\lvert {f(x) - f(x \pm h)} \right\rvert} < \varepsilon .\end{align*} These follow from the substitutions \(x_0-y = \mp h \implies y = x_0 \pm h\).

The main difference is that \(\delta\) may depend on \(x_0\) and \({\varepsilon}\) in continuity, but only depends on \({\varepsilon}\) in the uniform version. I.e. once \(\delta\) is fixed, for continuity one may only range over \(x\), but in uniform continuity one can range over all pairs \(x,y\).

Let \(X\) be a metric space and \(A\) a subset. Let \(A'\) denote the limit points of \(A\), and \(\mkern 1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern 1.5mu \coloneqq A\cup A'\) to be its closure.

  • A neighborhood of \(p\) is an open set \(U_p\) containing \(p\).

  • An \({\varepsilon}{\hbox{-}}\)neighborhood of \(p\) is an open ball \(B_r(p) \coloneqq\left\{{q {~\mathrel{\Big\vert}~}d(p, q) < r}\right\}\) for some \(r>0\).

  • A point \(p\in X\) is an accumulation point or a limit point of \(A\) iff every punctured neighborhood \(U_p\setminus\left\{{p}\right\}\) contains a point \(q\in A\), so \(q\neq p\).

  • If \(p\in A\) and \(p\) is not a limit point of \(A\), then \(p\) is an isolated point of \(A\).

  • \(A\) is closed iff \(A' \subset A\), so \(A\) contains all of its limit points.

  • A point \(p\in A\) is interior iff there is a neighborhood \(U_p \subset A\) that is strictly contained in \(A\).

  • \(A\) is open iff every point of \(A\) is interior.

  • \(A\) is perfect iff \(A\) is closed and \(A\subset A'\), so every point of \(A\) is a limit point of \(A\).

  • \(A\) is bounded iff there is a real number \(M\) and a point \(q\in X\) such that \(d(p, q) < M\) for all \(p\in A\).

  • \(A\) is dense in \(X\) iff every point \(x\in X\) is either a point of \(A\), so \(x\in A\), or a limit point of \(A\), so \(x\in A'\). I.e., \(X\subset A\cup A'\).

    • Alternatively, \(\mkern 1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern 1.5mu = X\), so the closure of \(A\) is \(X\).

A sequence of functions \(\left\{{ f_j }\right\}\) is said to converge pointwise to \(f\) if and only if \begin{align*} (\forall \varepsilon>0)(\forall x \in S)\left(\exists n_{0} = n_0(x, {\varepsilon}) \right)\left(\forall n>n_{0}\right)\left(\left|f_{n}(x)-f(x)\right|<\varepsilon\right) .\end{align*}

\begin{align*} (\forall \varepsilon>0)\left(\exists n_{0} = n_0({\varepsilon}) \right)(\forall x \in S)\left(\forall n>n_{0}\right)\left(\left|f_{n}(x)-f(x)\right|<\varepsilon\right) .\end{align*} Negated: 1 \begin{align*} (\exists \varepsilon>0)\left(\forall n_{0} = n_0 ({\varepsilon}) \right)(\exists x = x(n_0) \in S)\left(\exists n>n_{0}\right)\left(\left|f_{n}(x)-f(x)\right| \geq \varepsilon\right) .\end{align*}

Function Spaces

A metric space is complete if every Cauchy sequence converges.

If \(X\) is complete, then absolutely convergent implies convergent.

Recall that a set \(S\) in \(X\) is dense \(\iff\) every open \(U\subseteq X\) intersects \(S\). A set \(S\) is nowhere dense in \(X\) \(\iff\) the closure of \(S\) has empty interior \(\iff\) every subset (or interval) contains an open set (or a subinterval) that does not intersect \(S\). This just says \(S\) is not dense in any subset \(S' \subseteq X\), by negating what it means to be dense.

A set is meager if it is a countable union of nowhere dense sets.

A space \(X\) is a Baire space if and only if every countable intersections of open, dense sets is still dense.

Measure Theory

\begin{align*} \liminf_{n} E_{n} \coloneqq\displaystyle\bigcup_{N=1}^\infty \displaystyle\bigcap_{n=N}^\infty E_{n} &= \left\{{x {~\mathrel{\Big\vert}~}x\in E_{n} \text{ for all but finitely many } n}\right\} \\ \limsup_{n} E_{n} \coloneqq\displaystyle\bigcap_{N=1}^\infty \displaystyle\bigcup_{n=N}^{\infty} E_{n} &= \left\{{x {~\mathrel{\Big\vert}~}x\in E_{n} \text{ for infinitely many } n}\right\} .\end{align*}

How to derive these definitions: use that \(\inf\) corresponds to intersections/existence and \(\sup\) corresponds to unions/forall.

  • For \(\liminf E_n\):
    • \(x\in \liminf E_n \iff\) there exists some \(N\) such that \(x\in \cap_{n\geq N} E_n\), i.e. \(x\in E_n\) for all \(n\geq N\). So \(x\) is in all but finitely many \(n\).
    • How to remember: \(\liminf_{n} x_n = \sup_{n} \inf_{k\geq n} x_n\) for sequences, where sups look like unions and infs look like intersections.
    • Alternatively: there exists an \(n\) (union) such that for all \(k\geq n\) (intersection)…
  • For \(\limsup E_n\):
    • \(x\in \limsup E_n \iff\) for every \(N\), there exists some \(n\geq N\) such that \(x\in E_n\). So \(x\) is an infinitely many \(E_n\).
    • How to remember: \(\limsup_{n} x_n = \inf{n} \sup{k\geq n} x_n\) for sequences, where sups look like unions and infs look like intersections.
    • Alternatively: for all \(n\) (intersection) there exists a \(k\geq n\) (union)…

It’s also useful to note that \(\liminf E_n \subseteq \limsup E_n\), since \(\liminf E_n\) are elements that are eventually in all sets, and \(\limsup E_n\) are elements in infinitely many sets.

Why these are useful: for finite measure spaces, \begin{align*} \mu\qty{\liminf_n E_n }\leq \liminf_n \mu(E_n) \leq \lim_n \mu(E_n) \leq \limsup_n \mu(E_n) \leq \mu\qty{\limsup_n E_n} .\end{align*} If the \(\limsup\) and \(\liminf\) sets are equal, then one can define the set \(\lim_n E_n \coloneqq\cup_n E_n\) if \(E_n \nearrow E\) or \(\lim_n E_n \coloneqq\cap_n E_n\) if \(E_n\searrow E\) in which case continuity of measure states \begin{align*} \mu\qty{\lim_n E_n} = \lim_n \mu(E_n) .\end{align*}

An \(F_\sigma\) set is a union of closed sets, and a \(G_\delta\) set is an intersection of opens. 2

The outer measure of a set is given by \begin{align*} m_*(E) \coloneqq\inf_{\substack{\left\{{Q_{i}}\right\} \rightrightarrows E \\ \text{closed cubes}}} \sum {\left\lvert {Q_{i}} \right\rvert} ,\end{align*} where \({\left\lvert {Q_i} \right\rvert}\) is the standard Euclidean volume of a cube in \({\mathbb{R}}^n\).

A subset \(E\subseteq {\mathbb{R}}^n\) is Lebesgue measurable iff for every \({\varepsilon}> 0\) there exists an open set \(O \supseteq E\) such that \(m_*(O\setminus E) < {\varepsilon}\). In this case, we define \(m(E) \coloneqq m_*(E)\).

\(f\in L^+\) iff \(f\) is measurable and non-negative.

Integrals and \(L^p\) Spaces

A measurable function is integrable iff \({\left\lVert {f} \right\rVert}_1 < \infty\).

\begin{align*} {\left\lVert {f} \right\rVert}_\infty &\coloneqq\inf_{\alpha \geq 0} \left\{{\alpha {~\mathrel{\Big\vert}~}\mu\qty{\left\{{{\left\lvert {f} \right\rvert} \geq \alpha}\right\}} = 0}\right\} .\end{align*} In words, this is the smallest upper bound that holds almost everywhere, so \({\left\lvert {f(x)} \right\rvert} \leq {\left\lVert {f} \right\rVert}_\infty\) holds for almost every \(x\). A function \(f:X \to {\mathbb{C}}\) is essentially bounded iff there exists a real number \(c\) such that \(\mu(\left\{{{\left\lvert {f} \right\rvert} > x}\right\}) = 0\), i.e. \({\left\lVert {f} \right\rVert}_\infty < \infty\).

\begin{align*} L^\infty(X) \coloneqq\left\{{f: X\to {\mathbb{C}}{~\mathrel{\Big\vert}~}f \text{ is essentially bounded }}\right\} \coloneqq\left\{{f: X\to {\mathbb{C}}{~\mathrel{\Big\vert}~}{\left\lVert {f} \right\rVert}_{\infty }< \infty}\right\} .\end{align*}

\begin{align*}f * g(x)=\int f(x-y) g(y) d y .\end{align*}

\begin{align*} \widehat{f}(\xi) = \int f(x) ~e^{2\pi i x \cdot \xi} ~dx .\end{align*}

\begin{align*} \phi_{t}(x) = t^{-n} \phi\left(t^{-1} x\right) .\end{align*}

For \(\phi\in L^1\), the dilations satisfy \(\int \phi_{t} = \int \phi\), and if \(\int \phi = 1\) then \(\phi\) is an approximate identity.

Some properties that approximate identities enjoy:

  • \(\int \phi_t = {\left\lVert {\phi_t} \right\rVert}_1 = 1\) for every \(t\).
  • \(\sup_t {\left\lVert {\phi_t} \right\rVert}_1 < \infty\)
  • For every \(h>0\), \(\lim_{t\to\infty} \int_{{\left\lvert {x} \right\rvert} \geq h} {\left\lvert {\phi_t(x)} \right\rvert}\,dx= 0\).

Functional Analysis

For \(X\) a normed vector space and \(L \in X {}^{ \vee }\), the dual norm or operator norm is defined by \begin{align*} {\left\lVert {L} \right\rVert}_{X {}^{ \vee }} \coloneqq\sup_{ \substack{x\in X \\ {\left\lVert {x} \right\rVert} = 1} } {\left\lvert {L(x)} \right\rvert} = \sup_{ \substack{x\in X \\ {\left\lVert {x} \right\rVert} \leq 1} } {\left\lvert {L(x)} \right\rvert} .\end{align*}

A countable collection of elements \(\left\{{ u_i }\right\}\) is orthonormal if and only if

  • \({\left\langle {u_i},~{u_j} \right\rangle} = 0\) for all \(j \neq k\) and
  • \({\left\lVert {u_j} \right\rVert}^2 \coloneqq{\left\langle {u_j},~{u_j} \right\rangle} = 1\) for all \(j\).

A set \(\left\{{u_{n}}\right\}\) is a basis for a Hilbert space \({\mathcal{H}}\) iff it is dense in \({\mathcal{H}}\).

A collection of vectors \(\left\{{u_{n}}\right\}\subset H\) is complete iff \({\left\langle {x},~{u_{n}} \right\rangle} = 0\) for all \(n \iff x = 0\) in \(H\).

The dual of a Hilbert space \(H\) is defined as \begin{align*} H {}^{ \vee }\coloneqq\left\{{L: H\to {\mathbb{C}}{~\mathrel{\Big\vert}~}L \text{ is continuous }}\right\} .\end{align*}

A map \(L: X \to {\mathbb{C}}\) is a linear functional iff \begin{align*} L(\alpha\mathbf{x} + \mathbf{y}) = \alpha L(\mathbf{x}) + L(\mathbf{y}). .\end{align*}

A space is a Banach space if and only if it is a complete normed vector space.

A Hilbert space is an inner product space which is a Banach space under the induced norm.

Slogan: to negate, find a bad \(x\) depending on \(n_0\) that are larger than some \({\varepsilon}\).
Mnemonic: “F” stands for ferme, which is “closed” in French, and \(\sigma\) corresponds to a “sum”, i.e. a union.