I need to write norm of sum, but the sum symbol is larger than tho norm symbol (||) and it doesn't look good. Is there any symbol for norm which will adjust its size?
\documentclass[12pt,a4paper]{article}
\begin{document}
\begin{equation}
||\left(\sum_{n=1}^N \bf P_{\rm n}\rm\right) ||^2 = \left(\sum_n \frac{E_n}{c}\right)^2 - \left(\sum_n \bf p_{\rm n}\rm \right)^2
\end{equation}
\end{document}
Either of the following methods should work:
You could load the amsmath package and define a \norm macro as follows:
\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
and then write
\norm{ \biggl(\sum_{n=1}^N \mathbf{P}_{n}\biggr) }
in the equation of interest. Note that the round parentheses will be too big if you write \left( and \right); I recommend you write \biggl( and \biggr) instead.
Your example code may therefore be written as
\documentclass[12pt,a4paper]{article}
\usepackage{amsmath}
\newcommand\norm[1]{\left\lVert#1\right\rVert}
\begin{document}
\begin{equation}
\norm{ \biggl(\sum_{n=1}^N \mathbf{P}_{n}\biggr) }^2 =
\biggl(\sum_n \frac{E_n}{c}\biggr)^2 -
\biggl(\sum_n \mathbf{p}_{n} \biggr)^2
\end{equation}
\end{document}
Some might say that the resulting norm "fences" in the example above are a bit too large and thus threaten to dominate visually the rest of the math stuff. (This happens, of course, because the macro uses \left and \right directives to size the fences.) To fix this issue, you could load the mathtools package and insert the following instruction in the preamble:
\DeclarePairedDelimiterX{\norm}[1]{\lVert}{\rVert}{#1}
and use the \norm macro with an explicit size instruction (here: \bigg, i.e., the same size as for the parentheses):
\documentclass[12pt,a4paper]{article}
\usepackage{mathtools}
\DeclarePairedDelimiterX{\norm}[1]{\lVert}{\rVert}{#1}
\begin{document}
\begin{equation}
\norm[\bigg]{\biggl(\sum_{n=1}^N \mathbf{P}_{n}\biggr) }^2 =
\biggl(\sum_n \frac{E_n}{c}\biggr)^2 -
\biggl(\sum_n \mathbf{p}_{n} \biggr)^2
\end{equation}
\end{document}
\Vert is equivalent to \|. The l and r prefixes inform TeX that the commands \lVert and \rVert should be given math codes mathopen and mathclose, respectively. (TeX has no way of knowing how to modify the default math code of \Vert.)
\norm is not pre-defined in amsmath or some other package? It is a quite common operation.
\documentclass[12pt,a4paper]{article} with \documentclass{beamer}, the document still doesn't produce product symbols instead of summation symbols. All I can surmise is that one or more of the packages you're loading (I have no idea which one it could be... -- cf. my earlier comment) would appear to be messing with the meaning of \sum.
Use \Vert instead of ||. You can prefix \Vert with any brace-sizing-command, like \left, \right, \big, ..
\documentclass[12pt,a4paper]{article}
\begin{document}
\begin{equation}
\left\Vert\left(\sum_{n=1}^N \bf P_{\rm n}\rm\right) \right\Vert^2 = \left(\sum_n \frac{E_n}{c}\right)^2 - \left(\sum_n \bf p_{\rm n}\rm \right)^2
\end{equation}
\end{document}
absolute value. There are many questions here (like this one) which provide additional information, if you want to know more ;)
\rm commands used are wrong: besides being an obsolete command, the n should be in italics. Also \bf is obsolete, use \mathbf{P} and \mathbf{p}.
n in E_n in italics, so I think it was a choice to typeset the n in P_n in an upright shape.
\rm has been used in order to countermand \bf, but it's wrong to begin with.
\mathbf{P}_{n}and don't use\rmor\bfany more: they have been obsolete for more than ten years in LaTeX.