path: root/macros/latex/contrib/nchairx/source/chairxmathAnalysisDoc.dtx

% \iffalse meta-comment
%
% Copyright (C) 2018 - 2021 by ChairX
%
% This file may be distributed and/or modified under the
% conditions of the LaTeX Project Public License, either
% version 1.3 of this license or (at your option) any later
% version.  The latest version of this license is in:
%
%    http://www.latex-project.org/lppl.txt
%
% and version 1.3 or later is part of all distributions of
% LaTeX version 2005/12/01 or later.
%
% This file contains the documentation of all analysis related macros .
%
% Macros have to be described by (delete the first %)
% \DescribeMacro{\macro}
% Description and usage of the macro.
%
% The description will appear in the usage
% part of the documentation. Use \subsubsection{} etc. for structuring.
%
% The implementation of the macros defined here has to be written in
% chairxmathAnalysis.dtx
%\fi
%
%\subsubsection{General Anyalsis Macros} \label{sec:Doc_GeneralMacrosAnalysis}
%
% \DescribeMacro{\vol}
% Volume |\vol|: $\vol$\\
% Uses |operatorfont|
%
% \DescribeMacro{\complete}
% Completion of some space |\complete{V}|: $\complete{V}$
%
% \DescribeMacro{\Ball}
% Open ball |\Ball_{r}(p)|: $\Ball_{r}(p)$
%
% \DescribeMacro{\abs}
% Generic absolute value |\abs{x}|: $\abs{x}$
%
% \DescribeMacro{\norm}
% Generic norm |\norm{v}|: $\norm{v}$
%
% \DescribeMacro{\supnorm}
% Supremum norm |\supnorm{f}|: $\supnorm{f}$
%
% \DescribeMacro{\expands}
% Formal expansions |f(t) \stackrel{t \to 0}{\expands} t^k|:
% $f(t) \stackrel{t \to 0}{\expands} t^k$, \\
% or with optional stretching factor (default is 2.5) |a \expands[4] b|: $a \expands[4] b$.
%
%
% \subsubsection{Pseudodifferential Operators}
% \label{sec:Doc_PseudodifferentialOperators}
%
% \DescribeMacro{\std}
% Standard ordering as small subscript |\sigma_\std|: $\sigma_\std$ \\
% Uses |scriptfont|
%
% \DescribeMacro{\Weyl}
% Weyl ordering as small subscript |\sigma_\Weyl|: $\sigma_\Weyl$ \\
% Uses |scriptfont|
%
% \DescribeMacro{\Op}
% Operator for a symbol |\Op(f)|: $\Op(f)$ \\
% Uses |operatorfont|
%
% \DescribeMacro{\Opstd}
% Standard ordered operator for a symbol |\Opstd(f)|: $\Opstd(f)$ \\
% Uses |operatorfont|
%
% \DescribeMacro{\OpWeyl}
% Weyl ordered operator for a symbol |\OpWeyl(f)|: $\OpWeyl(f)$ \\
% Uses |operatorfont|
%
%
%\subsubsection{Function Spaces}
%
% \DescribeMacro{\spacename}
% Font for specific functional spaces |\spacename{F}(X)|: $\spacename{F}(X)$ \\
% Uses |spacefont|.
%
% \DescribeMacro{\Bounded}
% Bounded functions |\Bounded(X)|: $\Bounded(X)$\\
% Uses |spacefont|.
%
% \DescribeMacro{\Continuous}
% Continuous functions |\Continuous(X)|: $\Continuous(X)$\\
% Uses |spacefont|.
%
% \DescribeMacro{\Contbound}
% Continuous bounded functions |\Contbound(X)|: $\Contbound(X)$\\
% Uses |spacefont|.
%
% \DescribeMacro{\Fun}
% $C^k$-functions (for $\Continuous$ use |\Continuous|) |\Fun(M)|: $\Fun(M)$ and |\Fun[\ell](M)|: $\Fun[\ell](M)$ \\
% Uses |spacefont|.
%
% \DescribeMacro{\Cinfty}
% Smooth functions |\Cinfty|: $\Cinfty(M)$\\
% Uses |spacefont|.
%
% \DescribeMacro{\Comega}
% Real-analytic functions |\Comega|: $\Comega(M)$\\
% Uses |spacefont|.
%
% \DescribeMacro{\Holomorphic}
% Holomorphic functions |\Holomorphic|: $\Holomorphic(U)$\\
% Uses |spacefont|.
%
% \DescribeMacro{\AntiHolomorphic}
% Anti-holomorphic functions |\AntiHolomorphic|: $\AntiHolomorphic(U)$\\
% Uses |spacefont|.
%
% \DescribeMacro{\Schwartz}
% Schwartz space |\Schwartz|: $\Schwartz(\mathbb{R}^n)$\\
% Uses |spacefont|.
%
% \DescribeMacro{\Riemann}
% Riemann integrable functions |\Riemann([a, b])|: $\Riemann([a, b])$ \\
% Uses |spacefont|.
%
% \subsubsection{Locally Convex Analysis and Distributions}
%
% \DescribeMacro{\singsupp}
% Singular support of a distribution |\singsupp u|: $\singsupp u$
%
% \DescribeMacro{\seminorm}
% Font for generic seminorm |\seminorm{p}|: $\seminorm{p}$
%
% \DescribeMacro{\ord}
% Order of a distribution |\ord(u)|: $\ord(u)$
%
% \DescribeMacro{\conv}
% Convex hull |\conv(A)|: $\conv(A)$
%
% \DescribeMacro{\extreme}
% Extreme points |\extreme(A)|: $\extreme(A)$
%
% \subsubsection{Hilbert Spaces and Operators}
%
% \DescribeMacro{\hilbert}
% Font for Hilbert spaces |\hilbert{H}|: $\hilbert{H}$ \\
% Uses |hilbertfont|
%
% \DescribeMacro{\prehilb}
% Font for pre-Hilbert spaces |\prehilb{H}|: $\prehilb{H}$ \\
% Uses |prehilbfont|.
%
% \DescribeMacro{\Adjointable}
% Adjointable operators |\Adjointable(\hilbert{H})|:
% $\Adjointable(\hilbert{H})$ or with optional argument
% |\Adjointable[\algebra{A}](\hilbert{H})|: $\Adjointable[\algebra{A}](\hilbert{H})$ if
% we have a Hilbert module over an algebra  $\algebra{A}$ instead.
%
% \DescribeMacro{\Finite}
% Finite rank operators |\Finite(\hilbert{H})|: $\Finite(\hilbert{H})$
% or with optional argument
% |\Finite[\algebra{A}](\module{H})|: $\Finite[\algebra{A}](\module{H})$
%
% \DescribeMacro{\Compact}
% Compact operators |\Compact(\hilbert{H})|: $\Compact(\hilbert{H})$
% or with optional argument
% |\Compact[\algebra{A}](\module{H})|: $\Compact[\algebra{A}](\module{H})$
%
% \DescribeMacro{\opdomain}
% Domain of definition of an operator |\opdomain(A)|: $\opdomain(A)$ \\
% Uses |\hilbertfont|.
%
% \DescribeMacro{\spec}
% Spectrum of an operator |\spec(A)|: $\spec(A)$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\closure}
% Closure of an operator |\closure{A}|: $\closure{A}$
%
% \DescribeMacro{\res}
% Resolvent set of an operator |\res(A)|: $\res(A)$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\Res}
% Resolvent of an operator |\Res_z(A)|: $\Res_z(A)$\\
% Uses |operatorfont|.
%
% \DescribeMacro{\specrad}
% Spectral radius of an operator |\specrad(A)|: $\specrad(A)$
%
% \DescribeMacro{\slim}
% Strong limit |\slim_{n \longrightarrow \infty} A_n|: $\slim_{n \longrightarrow \infty} A_n$
%
% \DescribeMacro{\wlim}
% Weak limit |\wlim_{n \longrightarrow \infty} A_n|: $\wlim_{n \longrightarrow \infty} A_n$
%
% \subsubsection{Dirac's Bra and Ket Notation}
%
% \DescribeMacro{\bra}
% Dirac bra |\bra{\psi}|: $\bra{\psi}$
%
% \DescribeMacro{\ket}
% Dirac ket |\ket{\phi}|: $\ket{\phi}$
%
% \DescribeMacro{\braket}
% Dirac braket |\braket{\phi}{\psi}|: $\braket{\phi}{\psi}$
%
% \DescribeMacro{\ketbra}
% Dirac ketbra |\ketbra{\phi}{\psi}|: $\ketbra{\phi}{\psi}$
%
% \subsubsection{Operator Algebras}
%
% \DescribeMacro{\Spec}
% Spectrum of an algebra |\Spec(\algebra{A})|: $\Spec(\algebra{A})$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\Rad}
% Radical of an algebra |\Rad(\algebra{A})|: $\Rad(\algebra{A})$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\ind}
% Fredholm index (|\index| is already used!)  |\ind(A)|: $\ind(A)$  \\
% Uses |operatorfont|.
%
% \subsubsection{Measure Theory and Integration}
%
% Here we need various function space of integrable functions
% (calligraphic ones) and the corresponding quotients by zero
% functions (roman ones):
%
% \DescribeMacro{\Measurable}
% Measurable functions |\Measurable(X)|: $\Measurable(X)$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\Meas}
% Complex measures |\Meas(X)|: $\Meas(X)$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\BoundMeas}
% Bounded measurable functions |\BoundMeas(X)|: $\BoundMeas(X)$ \\
% Uses |spacefont|.
%
% \DescribeMacro{\Lp}
% Equivalence classes of $p$-integrable functions ($p$ is an optional
% argument) |\Lp(X)|: $\Lp(X)$ and |\Lp[q](X)|: $\Lp[q](X)$
%
% \DescribeMacro{\Lone}
% Equivalence classes of integrable functions |\Lone(X)|: $\Lone(X)$
%
% \DescribeMacro{\Ltwo}
%  Equivalence classes of square integrable functions |\Ltwo(X)|: $\Ltwo(X)$
%
%  \DescribeMacro{\Linfty}
%  Equivalence classes of essentially bounded functions |\Linfty(X)|: $\Linfty(X)$
%
%  \DescribeMacro{\Intp}
% Space of $p$-integrable functions |\Intp(X)|:
% $\Intp(X)$ and with optional argument
% |\Intp[q](X)|: $\Intp[q](X)$
%
% \DescribeMacro{\Intone}
% Space of integrable functions |\Intone(X)|: $\Intone(X)$
%
% \DescribeMacro{\Inttwo}
% Space of square integrable functions |\Inttwo(X)|: $\Inttwo(X)$
%
% \DescribeMacro{\Intinfty}
% Space of essentially bounded functions |\Intinfty(X)|: $\Intinfty(X)$
%
% \DescribeMacro{\essrange}
% Essential range |\essrange(f)|: $\essrange(f)$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\esssup}
% Essential supremum |\esssup(f)|: $\esssup(f)$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\esssupnorm}
% Essential supremum norm |\esssupnorm{f}|: $\esssupnorm{f}$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\ac}
% Absolutely continuous part of a measure |\mu_\ac|: $\mu_\ac$ \\
% Uses |scriptfont|.
%
% \DescribeMacro{\sing}
% Singular part of a measure |\mu_\sing|: $\mu_\sing$ \\
% Uses |scriptfont|.
%
% \subsubsection{Limits} \label{sec:Doc_Limits}
%
%\DescribeMacro{\indlim}
% Inductive (or direct) limit |\indlim_{i \in I} A_i|: $\indlim_{i \in I} A_i$\\
% Uses |operatorfont|.
%
%\DescribeMacro{\projlim}
% Projective (or inverse) limit |\projlim_{i \in I} A_i|: $\projlim_{i \in I} A_i$\\
% Uses |operatorfont|.