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definitions.tex
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definitions.tex
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\RequirePackage{mathrsfs} % Font \mathscr
\theoremstyle{plain}
\newtheorem{theorem}{Teorema}[section]
\newtheorem{proposition}[theorem]{Proposição}
\newtheorem{corollary}[theorem]{Corolário}
\newtheorem{lemma}[theorem]{Lema}
\newtheorem*{notation}{Notação}
\newtheorem{definition}[theorem]{Definição}
% Horrível: o estilo "definition" não é bom para definição e notação!
% Horrível: estilo "remark" é ruim para observações e exemplos!!
\theoremstyle{definition}
\newtheorem{example}[theorem]{Exemplo}
\newtheorem{obs}[theorem]{Observação}
% Commonly used sets.
\providecommand{\naturals}[0]{\mathbb{N}}
\providecommand{\integers}[0]{\mathbb{Z}}
\providecommand{\rationals}[0]{\mathbb{Q}}
\providecommand{\reals}[0]{\mathbb{R}}
\providecommand{\complexes}[0]{\mathbb{C}}
\providecommand{\family}[1]{{\mathscr{#1}}}
\providecommand{\parts}[1]{{\family{P}\left(#1\right)}}
% Special symbols.
\providecommand{\suchthat}[0]{{\thickspace | \thickspace}}
\providecommand{\cardinality}[1]{{\# #1}}
% Brackets.
\newcommand{\threebracket}[5]{{\left #1 \vphantom{{#2}^N}\vphantom{{#4}^N}{#2} \, \right #3 \left. \vphantom{{#2}^N}\vphantom{{#4}^N}{#4} \right #5}}
\providecommand{\setsuchthat}[2]{{\threebracket{\{}{#1}{|\:}{#2}{\}}}}
\providecommand{\set}[1]{{\left \{ {#1} \right \}}}
\providecommand{\abs}[1]{{\left\lvert#1\right\rvert}}
\providecommand{\norm}[2][]{{\left\lVert{#2}\right\rVert_{#1}}}
\newcommand{\integral}[3][]{{\int_{#1} {#2}\, \mathrm{d}{#3}}}
% Operator names.
\DeclareMathOperator{\identity}{id}
\DeclareMathOperator{\interiorop}{int}
\DeclareMathOperator{\closureop}{cl}
\DeclareMathOperator{\supportop}{supp}
\DeclareMathOperator{\qtpop}{qtp}
\DeclareMathOperator{\eigenspace}{eig}
\DeclareMathOperator{\expbase}{e}
% Functions
\newcommand{\function}[3]{{#1: #2 \to #3}}
\newcommand{\functionmaps}[3]{{#1: #2 \mapsto #3}}
\newcommand{\functionarray}[5]{{\begin{array}[t]{lrcl}
#1:& #2 &\to  \\
& #4 &\mapsto 
\end{array}}}
% Operations
\newcommand{\closure}[1]{{\overline{#1}}}
\newcommand{\interior}[1]{\interiorop \left({#1}\right)}
\newcommand{\complementset}[1]{{#1}^{c}}
\newcommand{\powerset}[1]{{\mathscr{P}({#1})}}
\newcommand{\indicator}[1]{{I_{#1}}}
\newcommand{\support}[1]{{\supportop \left( {#1} \right)}}
\newcommand{\ball}[3][]{{B_{#1}\left({#3}; {#2}\right)}}
\newcommand{\balls}[2]{{\family{B}_{#1}\left({#2}\right)}}