# Bracket for dual pair (in functional analysis)

In functional analysis, one occasionally considers a so-called dual pair (a pair of vector spaces tied together by a non-degenerate bilinear map to the ground field). I usually typeset these using \langle and \rangle, but it seems that all textbooks in functional analysis use a slightly different symbol for this. (See samples below.) The shape is very similar to \langle and \rangle, but the angle between the two legs is noticeably smaller. Consequently, the glyph is a bit wider.

How do I reproduce this symbol in LaTeX?

Samples:

1. LaTeX with \langle and \rangle:

2. H. H. Schaefer, with M. P. Wolff, Topological Vector Spaces, Second Edition (1999), p. 123:

3. Walter Rudin, Functional Analysis, Second Edition (1991), p. 93:

4. John B. Conway, A Course in Functional Analysis (1985), p. 127:

5. Gert K. Pedersen, Analysis Now, Corrected second printing (1995), p. 59:

6. Mahlon M. Day, Normed Linear Spaces, Second printing corrected (1962), p. 8:

... and the list goes on and on.

• I like the typography :-). For my low opinion peraphs the last solution (1962) not will have LaTeX the same symbols. Something of similar I think yes. – Sebastiano Jul 26 at 12:25
• I wouldn't worry; slightly different typographic choices don't affect the meaning. – egreg Jul 26 at 13:00

Here I can give you some examples.

1) with mtpro2, v. lite (no pay):

\documentclass[12pt]{article}
\usepackage[lite]{mtpro2}
%\usepackage{newtxtext,newtxmath}
\usepackage{braket}
\begin{document}
$\braket{F,G}$,
$\braket{x,x^{*}}$, $\braket{x,\varphi}$, $\braket{x,y}$
\end{document}


If you use the pro version you will get this:

2) with the use of newtxmath you will have the same characters of \usepackage[lite]{mtpro2}: https://www.pctex.com/mtpro2.html

\documentclass[12pt]{article}
\usepackage{newtxtext,newtxmath}
\usepackage{braket}
\begin{document}
$\braket{F,G}$,
$\braket{x,x^{*}}$, $\braket{x,\varphi}$, $\braket{x,y}$
\end{document}


3) Instead, using fourier option:

\documentclass[12pt]{article}
\usepackage{fourier}
\usepackage{braket}
\begin{document}
$\braket{F,G}$,
$\braket{x,x^{*}}$, $\braket{x,\varphi}$, $\braket{x,y}$
\end{document}


4) With mathpazo:

\documentclass[12pt]{article}
\usepackage{mathpazo}
\usepackage{braket}
\begin{document}
$\braket{F,G}$,
$\braket{x,x^{*}}$, $\braket{x,\varphi}$, $\braket{x,y}$
\end{document}


Into this link there is the LaTeX Font Catalogue with math support. https://tug.dk/FontCatalogue/mathfonts.html.

It is important to explain hat some of the mathematical fonts it is essential the use of XeLaTeX (https://it.overleaf.com/learn/latex/XeLaTeX) or LuaLaTeX (http://www.luatex.org/).

In the modern toolchain, with unicode-math, you can select a different math font of your choice. The list of symbols includes samples of a half-dozen different versions of \langle and \rangle. You might also take a look at \lcurvyangle and \rcurvyangle.

Here, for example, is the output with STIX Two Math, which you can also use in PDFLaTeX through the stix2 package.

It is also possible to mix-and-match symbols from different math fonts. Here is an example that uses only the angle brackets from STIX Two with the default math font, Latin Modern Math:

\documentclass{article}
\usepackage{mathtools}
\usepackage{unicode-math}

\setmathfont{Latin Modern Math}
\setmathfont[range={\langle,\rangle},
Scale=MatchUppercase
]{STIX Two Math}

\begin{document}
$$\langle x, \varphi \rangle$$
\end{document}


Flavor to taste. For example, Day’s book appears to be set in Palatino, which you might approximate with Asana Math or TeX Gyre Pagella Math today.

To fit the style of the default font, you can use the mathabx fonts. Here is how to use the langle, \rangle from mathabx without overwriting the default fonts. I added a \Braket macro, which uses the mathbx symbols, and has variable size if needed, thanks to the DecalrePairedDelimiter command from mathtools:

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amssymb}
\usepackage{mathtools}

\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}

\DeclareMathSymbol{\mylangle}{\mathopen}{matha}{"78}
\DeclareMathSymbol{\myrangle}{\mathclose}{matha}{"79}

\DeclareMathSymbol{\asymp}{\mathrel}{matha}{"16}
\DeclareMathSymbol{\notasymp}{\mathrel}{matha}{"1E}

\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}
\DeclareFontSubstitution{U}{mathx}{m}{n}
\DeclareMathDelimiter{\Langle}{\mathopen}{matha}{"78}
{mathx}{"40}
\DeclareMathDelimiter{\Rangle}{\mathclose}{matha}{"79}
{mathx}{"44}

\DeclarePairedDelimiter{\Braket}\Langle\Rangle

\begin{document}

$\Braket*{ x, \varphi }$%
$\langle x, \varphi \rangle$%
$\Braket[\big]{x, \varphi}$%

\end{document}