Absolute value symbol with spacing

Question

Often in math textbooks when they define an absolute value (or a norm in general) they use the symbol with a dot (even nothing) with some space inside. It looks like this:

Whatever method I use to define the absolute value symbol, normally I would only get a narrow spacing unless I define a new command for this special case. How do you do that?

Answer 1

Irrespective of how you generate the absolute-value and norm "fences" -- be it with | and \|, named macros such as \lvert/\rvert and \lVert/\rVert, or by defining bespoke macros called, say, \abs and \norm -- the adjustment that's required is always the same: surround \cdot with {}, aka "empty math atoms", i.e., write {}\cdot{}.

For the sake of ensuring notational consistency across the document -- a point made by @egreg in a comment -- it's a good idea not to write {}\cdot{} directly. Instead, define a macro called, say, \blank, in the preamble and then write \abs{\blank} and \norm{\blank} in the body of the document.

\documentclass{article}
\usepackage{mathtools} % for "[lr][vV]ert" and '\DeclarePairedDelimiter' macros
\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}
\DeclarePairedDelimiter{\norm}{\lVert}{\rVert}
\newcommand{\blank}{{}\cdot{}}
\begin{document}
$| \blank | \quad | \blank |$
$\lvert \blank \rvert \quad \lVert \blank \rVert$
$\abs{\blank} \quad \norm{\blank}$
\end{document}

Answer 2

You can do something like this.

\documentclass{article}
\usepackage{mathtools}
\newcommand*{\argordot}[1]{%
    \def\arg{#1}%
    \ifx\arg\empty
        \,\cdot\,%
    \else
        #1%
    \fi%
}
\DeclarePairedDelimiterX{\norm}[1]{\lVert}{\rVert}{\argordot{#1}}
\begin{document}
\(\norm{f}\) \(\norm{}\)
\end{document}