notations.tex

\newglossaryentry{not:def}{
  name={$A \coloneqq B$},
  description={define $A$ to equal $B$},
}

\newglossaryentry{not:set}{
  name={$\{ 1,2,3\}$},
  description={set with the given elements},
}

\newglossaryentry{not:equal}{
  name={$A = B$},
  description={$A$ and $B$ are equal},
}

\newglossaryentry{not:composition}{
  name={$f \circ g$},
  description={composition of functions},
}

\newglossaryentry{not:equivclass}{
  name={$[a]$},
  description={equivalence class of $a$},
}

\newglossaryentry{not:cardinality}{
  name={$\abs{A}$},
  description={cardinality of a set $A$},
}

\newglossaryentry{not:powerset}{
  name={$\sP(P)$},
  description={power set of $A$},
}

\newglossaryentry{not:emptyset}{
  name={$\emptyset$},
  description={the empty set},
}

\newglossaryentry{not:natnums}{
  name={$\N$},
  description={the natural numbers: $1,2,3,\ldots$},
}

\newglossaryentry{not:ints}{
  name={$\Z$},
  description={the integers: $\ldots,-2,-1,0,1,2,\ldots$},
}

\newglossaryentry{not:rationals}{
  name={$\Q$},
  description={the rational numbers},
}

\newglossaryentry{not:reals}{
  name={$\R$},
  description={the real numbers},
}

\newglossaryentry{not:complex}{
  name={$\C$},
  description={the complex numbers},
}

\newglossaryentry{not:euclidspace}{
  name={$\R^n$},
  description={the $n$-dimensional euclidean space},
}

\newglossaryentry{not:func}{
  name={$f \colon A \to B$},
  description={function with domain $A$ and codomain $B$},
}

\newglossaryentry{not:dirimage}{
  name={$f(A)$},
  description={direct image of $A$ by $f$},
}

\newglossaryentry{not:invimage}{
  name={$f^{-1}(A)$},
  description={inverse image of $A$ by $f$},
}

\newglossaryentry{not:invfunction}{
  name={$f^{-1}$},
  description={inverse function},
}

\newglossaryentry{not:abs}{
  name={$\abs{x}$},
  description={absolute value},
}

\newglossaryentry{not:closedbdinterval}{
  name={$[a,b]$},
  description={closed bounded interval},
}

\newglossaryentry{not:openbdinterval}{
  name={$(a,b)$},
  description={open bounded interval},
}

\newglossaryentry{not:halfopenbdinterval}{
  name={$(a,b]$, $[a,b)$},
  description={half-open bounded interval},
}

\newglossaryentry{not:closedubdinterval}{
  name={$[a,\infty)$, $(-\infty,b]$},
  description={closed unbounded interval},
}
\newglossaryentry{not:openubdinterval}{
  name={$(a,\infty)$, $(-\infty,b)$},
  description={open unbounded interval},
}

\newglossaryentry{not:segmentinRn}{
  name={$[x,y]$},
  description={line segment},
}

\newglossaryentry{not:sequence}{
  name={$\{x_n\}_{n=1}^\infty$},
  description={sequence},
}

\newglossaryentry{not:subsequence}{
  name={$\{x_{n_i}\}_{i=1}^\infty$},
  description={subsequence},
}

\newglossaryentry{not:sup}{
  name={$\sup\,E$},
  description={supremum of $E$},
}

\newglossaryentry{not:inf}{
  name={$\inf\,E$},
  description={infimum of $E$},
}

\newglossaryentry{not:min}{
  name={$\min\,E$},
  description={minimum of $E$},
}

\newglossaryentry{not:max}{
  name={$\max\,E$},
  description={maximum of $E$},
}

\newglossaryentry{not:supf}{
  name={$\displaystyle \sup_{x\in D} f(x)$},
  description={supremum of $f(D)$},
}

\newglossaryentry{not:inff}{
  name={$\displaystyle \inf_{x\in D} f(x)$},
  description={infimum of $f(D)$},
}

\newglossaryentry{not:extreal}{
  name={$\R^*$},
  description={the extended real numbers},
}

\newglossaryentry{not:infinity}{
  name={$\infty$},
  description={infinity},
}

\newglossaryentry{not:limseq}{
  name={$\displaystyle \lim_{n\to\infty} x_n$},
  description={limit of a sequence},
}

\newglossaryentry{not:limfunc}{
  name={$\displaystyle \lim_{x \to c} f(x)$},
  description={limit of a function},
}

\newglossaryentry{not:limitasarrows}{
  name={$f(x) \to L$ as $x \to c$},
  description={$f(x)$ converges to $L$ as $x$ goes to $c$},
}

\newglossaryentry{not:limfuncinf}{
  name={$\displaystyle \lim_{x \to \infty} f(x)$,
$\displaystyle \lim_{x \to -\infty} f(x)$},
  description={limit of a function at infinity},
}

\newglossaryentry{not:onesidedlim}{
  name={$\displaystyle \lim_{x \to c^{+}} f(x)$,
$\displaystyle \lim_{x \to c^{-}} f(x)$},
  description={one-sided limit of a function},
}

\newglossaryentry{not:limsupseq}{
  name={$\displaystyle \limsup_{n\to\infty} x_n$},
  description={limit superior},
}

\newglossaryentry{not:liminfseq}{
  name={$\displaystyle \liminf_{n\to\infty} x_n$},
  description={limit inferior},
}

\newglossaryentry{not:finitesum}{
  name={$\displaystyle \sum_{n=1}^k a_n$},
  description={sum $a_1 + a_2 + \cdots+ a_k$},
}

\newglossaryentry{not:series}{
  name={$\displaystyle \sum_{n=1}^\infty a_n$},
  description={series},
}

\newglossaryentry{not:riemannint}{
  name={$\displaystyle \int_a^b f$, $\displaystyle \int_a^b f(x) \, dx$},
  description={Riemann integral of $f$ on $[a,b]$},
}

\newglossaryentry{not:integrablefunc}{
  name={$\sR\bigl([a,b]\bigr)$},
  description={Riemann integrable functions on $[a,b]$},
}

\newglossaryentry{not:setbuilder}{
  name={$\{ x \in S : P(x) \}$},
  description={set building notation},
}

\newglossaryentry{not:subset}{
  name={$A \subset B$},
  description={$A$ is a subset of $B$},
}

\newglossaryentry{not:subsetneq}{
  name={$A \subsetneq B$},
  description={$A$ is a proper subset of $B$},
}

\newglossaryentry{not:eltof}{
  name={$x \in S$},
  description={$x$ is an element of $S$},
}
\newglossaryentry{not:noteltof}{
  name={$x \notin S$},
  description={$x$ is not an element of $S$},
}

\newglossaryentry{not:intersection}{
  name={$A \cap B$},
  description={intersection of $A$ and $B$},
}

\newglossaryentry{not:setminus}{
  name={$A \setminus B$},
  description={set minus, elements of $A$ not in $B$},
}

\newglossaryentry{not:complement}{
  name={$B^c$},
  description={set complement, elements not in $B$},
}

\newglossaryentry{not:cartprod}{
  name={$A \times B$},
  description={Cartesian product of $A$ and $B$},
}

\newglossaryentry{not:union}{
  name={$A \cup B$},
  description={union of $A$ and $B$},
}

\newglossaryentry{not:infunion}{
  name={$\displaystyle \bigcup_{n=1}^\infty A_n$},
  description={union of all $A_n$ for all $n \in \N$},
}

\newglossaryentry{not:arbitunion}{
  name={$\displaystyle \bigcup_{\lambda \in I} A_\lambda$},
  description={union of all $A_\lambda$ for all $\lambda \in I$},
}

\newglossaryentry{not:infintersection}{
  name={$\displaystyle \bigcap_{n=1}^\infty A_n$},
  description={intersection of all $A_n$ for all $n \in \N$},
}

\newglossaryentry{not:arbitintersection}{
  name={$\displaystyle \bigcup_{\lambda \in I} A_\lambda$},
  description={intersection of all $A_\lambda$ for all $\lambda \in I$},
}

\newglossaryentry{not:lowerdarbouxsum}{
  name={$L(P,f)$},
  description={lower Darboux sum of $f$ over partition $P$},
}

\newglossaryentry{not:upperdarbouxsum}{
  name={$U(P,f)$},
  description={upper Darboux sum of $f$ over partition $P$},
}

\newglossaryentry{not:lowerdarboux}{
  name={$\displaystyle \underline{\int_a^b}f$},
  description={lower Darboux integral},
}

\newglossaryentry{not:upperdarboux}{
  name={$\displaystyle \overline{\int_a^b}f$},
  description={upper Darboux integral},
}

\newglossaryentry{not:exp}{
  name={$\exp(x)$, $e^x$},
  description={exponential function},
}

\newglossaryentry{not:ln}{
  name={$\ln(x)$, $\log(x)$},
  description={natural logarithm function},
}

\newglossaryentry{not:pow}{
  name={$x^y$},
  description={exponentiation of $x > 0$ and $y \in \R$},
}

\newglossaryentry{not:e}{
  name={$e$},
  description={Euler's number, base of the natural logarithm},
}

\newglossaryentry{not:contfuncs}{
  name={$C(S,\R)$},
  description={continuous functions $f \colon S \to \R$},
}

\newglossaryentry{not:contdifffuncs}{
  name={$C^1(S,\R)$},
  description={continuously differentiable functions $f \colon S \to \R$},
}

\newglossaryentry{not:derivative}{
  name={$f'(x)$, $\frac{df}{dx}$, $\frac{d}{dx}\bigl( f(x) \bigr)$},
  description={derivative of $f$},
}

\newglossaryentry{not:secondthirdfourthder}{
  name={$f''$, $f'''$, $f''''$},
  description={second, third, fourth derivative of $f$},
}

\newglossaryentry{not:nthder}{
  name={$f^{(n)}$},
  description={$n$th derivative of $f$},
}

\newglossaryentry{not:ordereq}{
  name={$x = y$},
  description={$x$ is equal to $y$},
}

\newglossaryentry{not:orderlt}{
  name={$x < y$},
  description={$x$ is less than $y$},
}

\newglossaryentry{not:orderleq}{
  name={$x \leq y$},
  description={$x$ is less than or equal to $y$},
}

\newglossaryentry{not:ordergt}{
  name={$x > y$},
  description={$x$ is greater than $y$},
}


\newglossaryentry{not:ordergeq}{
  name={$x \geq y$},
  description={$x$ is greater than or equal to $y$},
}

\newglossaryentry{not:uniformnorm}{
  name={$\norm{f}_S$},
  description={uniform norm of $f$ over $S$},
}

\newglossaryentry{not:diam}{
  name={$\operatorname{diam}(S)$},
  description={diameter of $S$},
}

\newglossaryentry{not:openball}{
  name={$B(p,\delta)$, $B_X(p,\delta)$},
  description={open ball in a metric space},
}

\newglossaryentry{not:closedball}{
  name={$C(p,\delta)$, $C_X(p,\delta)$},
  description={closed ball in a metric space},
}

\newglossaryentry{not:closure}{
  %FIXME: this is weird, we need protect
  name={$\protect\widebar{A}$},
  description={closure of $A$},
}

\newglossaryentry{not:interior}{
  name={$A^\circ$},
  description={interior of $A$},
}

\newglossaryentry{not:boundary}{
  name={$\partial A$},
  description={boundary of $A$},
}

\newglossaryentry{not:vector}{
  name={$(v_1,v_2,\ldots,v_n)$},
  description={vector},
}

\newglossaryentry{not:realpoly}{
  name={$\R[t]$},
  description={the set of polynomials in $t$},
}

\newglossaryentry{not:span}{
  name={$\spn(Y)$},
  description={span of the set $Y$},
}

\newglossaryentry{not:standardbasis}{
  name={$e_j$},
  description={standard basis vector $(0,\ldots,0,1,0,\ldots,0)$},
}

\newglossaryentry{not:LXY}{
  name={$L(X,Y)$},
  description={set of linear maps from $X$ to $Y$},
}

\newglossaryentry{not:LX}{
  name={$L(X)$},
  description={set of linear operators on $X$},
}

\newglossaryentry{not:dotprod}{
  name={$x \cdot y$},
  description={dot product of $x$ and $y$},
}

\newglossaryentry{not:norm}{
  name={$\snorm{\cdot}$},
  description={norm on a vector space},
}

\newglossaryentry{not:normRn}{
  name={$\snorm{\cdot}_{\R^n}$},
  description={the euclidean norm on $\R^n$},
}

\newglossaryentry{not:normLXY}{
  name={$\snorm{\cdot}_{L(X,Y)}$},
  description={operator norm on $L(X,Y)$},
}

\newglossaryentry{not:sgn}{
  name={$\operatorname{sgn}(x)$},
  description={sign function},
}

\newglossaryentry{not:det}{
  name={$\det(A)$},
  description={determinant of $A$},
}

\newglossaryentry{not:mvder}{
  name={$f'$, $Df$},
  description={derivative of $f$},
}

\newglossaryentry{not:partialder}{
  name={$\frac{\partial f}{\partial x_j}$}, %, $D_j f$},
  description={partial derivative of $f$ with respect to $x_j$},
}

\newglossaryentry{not:gradient}{
  name={$\nabla f$},
  description={gradient of $f$},
}

\newglossaryentry{not:mvdirder}{
  name={$D_u f$, $\frac{\partial f}{\partial u}$},
  description={directional derivative of $f$},
}

\newglossaryentry{not:jacobdet}{
  name={$J_f$,
$\frac{\partial(f_1,f_2,\ldots,f_n)}{\partial(x_1,x_2,\ldots,x_n)}$},
  description={Jacobian determinant of $f$},
}

\newglossaryentry{not:C1}{
  name={$C^1$, $C^1(U)$},
  description={continuously differentiable function/mapping},
}

\newcommand{\notsamplevectorcol}{$\left[
\begin{smallmatrix}v_1\\\vdots\\v_n\end{smallmatrix} \right]$}
\newglossaryentry{not:vectorcol}{
  name={\protect\notsamplevectorcol},
  description={vector (column vector)},
}

\newcommand{\notsamplematrix}{$\left[
\begin{smallmatrix}
a_{1,1} & \cdots & a_{1,n}\\
\vdots & \ddots & \vdots \\
a_{m,1} & \cdots & a_{m,n}
\end{smallmatrix} \right]$}
\newglossaryentry{not:matrix}{
  name={\protect\notsamplematrix},
  description={matrix},
}

\newglossaryentry{not:Ck}{
  name={$C^k$},
  description={$k$-times continuously differentiable function},
}

\newglossaryentry{not:subpartder}{
  name={$f_{x_1x_2}$},
  description={derivative of $f$ with respect to $x_1$ and then $x_2$},
}

\newglossaryentry{not:multivarpartder}{
  name={$\frac{\partial^2 f}{\partial x_2 \partial x_1}$},
  description={derivative of $f$ with respect to $x_1$ and then $x_2$},
}

\newglossaryentry{not:oneform}{
  name={$\omega_1 \,dx_1 + \omega_2 \,dx_2 + \cdots + \omega_n \,dx_n$},
  description={differential one-form},
}

\newglossaryentry{not:pathintegralomega}{
  name={$\int_{\gamma} \omega$},
  description={path integral of a one-form},
}

\newglossaryentry{not:lineintegralf}{
  name={$\int_{\gamma} f \,ds$, $\int_{\gamma} f(x) \,ds(x)$},
  description={line integral of $f$ against arc-length measure},
}

\newglossaryentry{not:pathintegralvecfield}{
  name={$\int_{\gamma} v \cdot d\gamma$},
  description={path integral of a vector field},
}

\newglossaryentry{not:ndimvolume}{
  name={$V(R)$},
  description={$n$-dimensional volume},
}

\newglossaryentry{not:lowerdarbouxR}{
  name={$\displaystyle \underline{\int_R}f$},
  description={lower Darboux integral over rectangle $R$},
}

\newglossaryentry{not:upperdarbouxR}{
  name={$\displaystyle \overline{\int_R}f$},
  description={upper Darboux integral over rectangle $R$},
}

\newglossaryentry{not:riemannintR}{
  name={$\displaystyle \int_R f$, $\displaystyle \int_R f(x) \, dx$, $\displaystyle \int_R f(x) \, dV$},
  description={Riemann integral of $f$ on $R$},
}

\newglossaryentry{not:integrablefuncR}{
  name={$\sR(R)$},
  description={Riemann integrable functions on $R$},
}

\newglossaryentry{not:outermeasure}{
  name={$m^*(S)$},
  description={outer measure of $S$},
}

\newglossaryentry{not:indicatorfunction}{
  name={$\chi_S$},
  description={indicator function of $S$},
}

\newglossaryentry{not:prod}{
  name={$\prod$},
  description={product},
}

\newglossaryentry{not:mapsto}{
  name={$x \mapsto y$},
  description={function that takes $x$ to $y$},
}

\newglossaryentry{not:GLX}{
  name={$GL(X)$},
  description={invertible linear operators on $X$},
}

\newglossaryentry{not:imaginary}{
  name={$i$},
  description={the imaginary number, $\sqrt{-1}$},
}

\newglossaryentry{not:conj}{
  %FIXME: this is weird, we need protect
  name={$\protect\widebar{z}$},
  description={complex conjugate of $z$},
}

\newglossaryentry{not:modulus}{
  name={$\sabs{z}$},
  description={modulus of $z$},
}

\newglossaryentry{not:realpart}{
  name={$\Re\, z$},
  description={real part of $z$},
}

\newglossaryentry{not:imagpart}{
  name={$\Im\, z$},
  description={imaginary part of $z$},
}

\newglossaryentry{not:complexexp}{
  name={$e^z$},
  description={complex exponential function},
}

\newglossaryentry{not:sin}{
  name={$\sin(z)$},
  description={sine function},
}

\newglossaryentry{not:cos}{
  name={$\cos(z)$},
  description={cosine function},
}

\newglossaryentry{not:pi}{
  name={$\pi$},
  description={the number $\pi$},
}

\newglossaryentry{not:FS}{
  name={$\displaystyle f(x) \sim \sum_{n=-\infty}^\infty c_n e^{inx}$},
  description={Fourier series for $f$},
}

\newglossaryentry{not:L2innprod}{
  name={$\langle f, g \rangle$},
  description={inner product of functions},
}

\newglossaryentry{not:L2norm}{
  name={$\snorm{f}_2$},
  description={$L^2$ norm of $f$},
}

\newglossaryentry{not:FSsympartsum}{
  name={$s_N(f;x)$},
  description={symmetric partial sum of a Fourier series},
}


\newglossaryentry{not:oscillation}{
  name={$o(f,x,\delta)$, $o(f,x)$},
  description={oscillation of a function at $x$},
}