Theorems

Note: Most of the tips below require the amslatex macros. These are automatically loaded if you use the "amsart" documentclass, but if you use other documentclasses, such as "article", put the following in the preamble, after "\documentclass{...}":
\usepackage{amsmath, amsthm, amssymb}
For more about the amslatex macros, and as a general reference for the tips below, see Gratzer's "Math into LaTeX".

Disabling automatic theorem numbering

By default, LaTeX numbers theorems automatically. While in most situations, this is desirable, there are occasions where one doesn't want a theorem to be numbered; for example, a common practice is to use a different labeling scheme for theorems that are quoted from the literature, e.g., ``Theorem A'', or ``Theorem (Gauss)'', or ``Dirichlet's Theorem''.

Some authors try to deal with this problem by manually coding theorems, etc., resorting to constructs like

\vspace{5mm}\noindent{\bf Theorem A}. {\it We have ...}\vspace{5mm}
However, such "manual formatting" is bad LaTeX practice, should be avoided. For one, the spacing is likely not optimal, and it would take some trial and error to optimize the spacing. However, any such effort is most likely wasted, since such manual formatting will likely be removed at the publisher's end and replaced by proper LaTeX constructs.

The right way to handle situations like this is to use the "\newtheorem*" command, which works just like "\newtheorem", except that it doesn't number the theorems: To get the above example, add "\newtheorem*{thma}{Theorem A}" to the preamble (this defines a theorem, "thma"), and then use

\begin{thma} We have ... \end{thma}
The "\newtheorem*" command is part of the amslatex macro set; specifically, it is defined in the package "amsthm", which should be loaded (along with "amsmath" and "amssymb") as recommended above, unless you use "amsart" as documentclass (since the "amsart" class loads these packages automatically).

Automatic theorem numbering

Theorems declared with the "\newtheorem" command are automatically numbered. The numbering scheme is highly customizable (see Gratzer's book for details). The LaTeX default is to number each "theorem" consecutively: Theorem 1, Theorem 2, Theorem 3, Corollary 1, Corollary 2, Lemma 1, etc. However, except for short papers with only a few "theorems", this is usually not a good practice. A better scheme, that is appropriate for most papers and could be used in article templates, is one that numbers all theorems (and their equivalents) consecutively, but within each section: Theorem 1.1, Corollary 1.2, Lemma 1.3, etc. This is achieved with the following commands in the preamble:
\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
Then theorems, corollaries, and lemmas can be called up (and get automatically numbered) with
\begin{thm}\label{...} .... \end{thm}
\begin{cor}\label{...}  .... \end{cor}
\begin{lem}\label{...}  .... \end{lem}
where "\label{...}" contains the theorem label. The label is not mandatory, but needed if you want to refer back to the theorem with "\ref{...}".

Theorem styles

The "amsthm" package (loaded via the "amsart" documentclass, or explicitly with a "\usepackage{...}" command) provides different "theorem styles", so that theorems, corollaries, and lemmas, can be differentiated from definitions and remarks. In mathematical typesetting, it is customary to set the latter two (definitions and remarks) in ordinary Roman font, while the "real" theorems are set in italics. To achieve this effect, precede the theorem declarations corresponding to definitions and remarks by an appropriate "\theoremstyle{...}" command, as in the following example.
\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}

\theoremstyle{remark}
\newtheorem{rem}[thm]{Remark}

\theoremstyle{definition}
\newtheorem{def}[thm]{Definition}
Note that the "real" theorems (thm, cor,lem) need no explicit theoremstyle declaration, since the style appropriate for these theorems is the default. For more on this see Gratzer's book.

Proofs

For proofs use the "\begin{proof} ... \end{proof}" environment, which acts much like the "\demo ... \enddemo" pair in amstex. Here are some hints:

Other hints


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Last modified: Mon 29 Aug 2011 01:17:14 PM CDT A.J. Hildebrand