How can I make a scaling matrix norm symbol in overleaf with three bars?

Question

How can I write |||A||| with three bars in equal distances that scale? I read through the exiting entries and none of the solutions worked for me, the closest one was with "vertiii" but this put out bars in different sizes.

Answer 1

Using barbara's approach at Extensible double angle, etc, which essentially uses triple nested \left| and likewise \right|, with kerning added.

\documentclass{article}

\usepackage{mathtools}

\makeatletter
\def\@tvsp{\mathchoice{{}\mkern-4.5mu}{{}\mkern-4.5mu}{{}\mkern-2.5mu}{}}
\def\ltrivert{\left|\@tvsp\left|\@tvsp\left|}
\def\rtrivert{\right|\@tvsp\right|\@tvsp\right|}
\makeatother
\newcommand\tnorm[1]{\ltrivert#1\rtrivert}


\begin{document}

\[
\tnorm{x}  \tnorm{ \frac{A}{B} }
\]
\end{document}

Answer 2

The mathabx package defines a \vvvert delimiter. Here is a code to use it without loading the package, based on \DeclarePairedDelimiterX from mathtools:

\documentclass{article}
\usepackage{mathtools}
\usepackage{etoolbox} 

\DeclareFontFamily{U}{matha}{\hyphenchar\font45}
\DeclareFontShape{U}{matha}{m}{n}{
<-6> matha5 <6-7> matha6 <7-8> matha7
<8-9> matha8 <9-10> matha9
<10-12> matha10 <12-> matha12
}{}
\DeclareSymbolFont{matha}{U}{matha}{m}{n}

\DeclareFontFamily{U}{mathx}{\hyphenchar\font45}
\DeclareFontShape{U}{mathx}{m}{n}{
<-6> mathx5 <6-7> mathx6 <7-8> mathx7
<8-9> mathx8 <9-10> mathx9
<10-12> mathx10 <12-> mathx12
}{}
\DeclareSymbolFont{mathx}{U}{mathx}{m}{n}

\DeclareMathDelimiter{\vvvert} {0}{matha}{"7E}{mathx}{"17}%

\DeclarePairedDelimiterX{\normiii}[1]
{\vvvert}
{\vvvert}
{\ifblank{#1}{\: · \:}{#1}}

\begin{document}

\[ \normiii[\Big]{A^k } \qquad \normiii{} \]

\[ \normiii*{∑_{k \geq 0}^∞ A_k} \]

\end{document}