Why are the norm symbols || not aligned in this situation?

Question

I have a following code to align a system of equations:

\begin{aligned}
&  \|\partial_{j_1} \cdots \partial_{j_m} f (a)- \partial_{j_1} \cdots \partial_{j_m} f (x) \|\\
={}&  \| \partial^m f (a) \left  [e_{j_1}, \ldots, e_{j_m}\right ] - \partial^m f (x) \left  [e_{j_1}, \ldots, e_{j_m}\right ]\| \\
={}&  \left \| \big ( \partial^m f (a) - \partial^m f (x) \big) \left  [e_{j_1}, \ldots, e_{j_m}\right ] \right \| \\
\le{}&  \| \partial^m f (a) - \partial^m f (x)\| \cdot \| e_{j_1} \| \cdots \| e_{j_m} \| \\
={}&  \| \partial^m f (a) - \partial^m f (x)\|
\end{aligned}

with the result

Could you please explain why the norm symbols || (the third one) are not aligned in this situation? Thank you so much!

Answer 1

I think there are two issues with your code

• \left and \right can insert whitespace. Don't use it here; or, load the mleftright package and use \mleft and \mright instead of \left and \right.

• An unusual (to put it mildly) usage of the & alignment points.

Once these issues are fixed, the result is what one would expect it to be. To improve code legibility, I would also define a macro called \norm, using the \DeclarePairedDelimiter macro of the mathtools package. (The mathtools package is a superset of the amsmath package.)

\documentclass{article}
\usepackage{mathtools}
\DeclarePairedDelimiter{\norm}{\lVert}{\rVert}

\begin{document}
$\begin{aligned}
&\mathrel{\phantom{=}} 
     \norm{ \partial_{j_1} \dotsb \partial_{j_m} f(a) - 
            \partial_{j_1} \dotsb \partial_{j_m} f(x) } \\
&=   \norm{ \partial^m f(a) [e_{j_1}, \dots, e_{j_m} ] - 
            \partial^m f(x) [e_{j_1}, \dots, e_{j_m} ] } \\
&=   \norm[\big]{ 
            \bigl( \partial^m f(a) - \partial^m f(x) \bigr) 
                 [ e_{j_1}, \dots, e_{j_m} ] } \\
&\le \norm{ \partial^m f(a) - \partial^m f(x)} \cdot
     \norm{ e_{j_1}} \dotsb \norm{e_{j_m} } \\
&=   \norm{ \partial^m f(a) - \partial^m f(x)}
\end{aligned}$
\end{document}