I'm pretty sure it's not closed under vector addition but verification would be appreciated.
Is the set $S = \{\vec{x} \in \mathbb{R}^n : \|\vec{x}\| = |x_1|\}$ a subspace of $\mathbb{R}^n$?
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linear-algebra
vector-spaces
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0If by $x1$ you mean the first coordinate of $x$, then the set consists actually of those vectors whose all other coordinates except possibly the first are zero. – 2011-10-18
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1Look at the case $n=2$ for concreteness. Look at the vector $(a,b)$. To say its norm is $|a|$ is to say that $a^2+b^2=a^2$, so it's a fancy way of saying $b=0$. For sure it is a subspace! Argument for general $n$ is the same. – 2011-10-18