07-notation.tex

\chapter*{Notation}

\section*{Sets and symbols}

\begin{itemize}
    \item $\R$, $\R^+$ and $\R^-$ denote the sets of: all, positive (including $0$), and negative (including $0$) real numbers respectively.
    \item $\Z$ denotes the set of all integers,
        while $\N$ contains all positive integers including $0$.
    \item $\Kronecker i j$ is the \emph{Kronecker delta};
        it is equal to $1$ if $i = j$,
        and $0$ otherwise.
    \item $\grad$ denotes the \emph{gradient} on the vector space $\VectorSpace$.
    \item $(\LieAlgebra, \LieBracket \dummy \dummy)$ is the \emph{Lie algebra} of $\Group$.
    \item $\Laplacian$ denotes the \emph{Laplace operator} on $\Group$.
    \item $\adj T$ denotes the \emph{adjoint} of $T$.
    \item $e_\Group$ is the identity element of a group $\Group$.
    \item $e_1, \dots, e_{\dim \VectorSpace}$ denotes an orthonormal basis of $\VectorSpace$.
    \item $\conv f g$ represents the convolution of $f$ and $g$.
    \item $\delta_g$ is the delta-distribution at $g \in \Group$.
    \item $\Schwartz \Group$ is the set of \emph{Schwartz functions} on $\Group$.
    \item $\TemperedDistributions \Group$ is the set of \emph{tempered distributions} on $\Group$.
    \item $\SmoothFunctions \Group$ denotes the set of \emph{smooth functions} on $\Group$.
    \item $\Lebesgue p \Group$ will denote the set of all $\mu$-measurable complex functions $f$ on $\Group$ such that $\abs f^p$ is $\mu$-integrable;
        here, $\mu$ is a Haar measure on $\Group$.
    \item $\Sobolev s$ is the $2$-\emph{Sobolev space} of order $s$ on $\Group$.
    \item $\SobolevOrder p s$ is the $p$-\emph{Sobolev space} of order $s$ on $\Group$.
    \item $\Lin {\mathcal H_1, \mathcal H_2}$ denotes a bounded linear map between the Hilbert spaces $\mathcal H_1$ and $\mathcal H_2$; also, we write $\Lin {\mathcal H} \defeq \Lin {\mathcal H, \mathcal H}$.
    \item $\SchattenClasses p {\mathcal H}$ denotes the $p$-\emph{Schatten classes} on the Hilbert space $\mathcal H$.
    \item $\dualGroup \Group$ denotes the \emph{unitary dual} of $\Group$.
    \item $\Fourier$ denotes the \emph{(unitary) Fourier transform} on $\Group$.
    \item $\InverseFourier$ denotes the \emph{inverse Fourier transform} on $\Group$.
    \item $\JapaneseBracket \Group \dummy$ denotes the \emph{Japanese bracket}.
    \item $\SymbolClass m {\rho, \delta}$ denotes the class of symbols with order $m$ and type $(\rho, \delta)$ on $\Group$.
    \item $\OperatorClass m {\rho, \delta}$ denotes the class of operators with order $m$ and type $(\rho, \delta)$ on $\Group$.
\end{itemize}