I don't think you really wants this. However, here it is:
\documentclass{article}\usepackage{amsmath}\newcommand{\xnorm}[1]{% \mathchoice{\Bigl\|#1\Bigr\|}{\lVert#1\rVert}{\lVert#1\rVert}{\lVert#1\rVert}%}\begin{document}A norm in text mode is $\xnorm{v}$, while in display math it is\[\xnorm{v}=\sum_{i=1}^n\lvert v_i\rvert\]\end{document}The result is really ugly (and wrong):
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If you change \Bigl and \Bigr with \bigl and \bigr you get a more acceptable result, but I can't see the point of increasing the fences' size.
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