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So, given that $$W = \{[x, y] \mid y = |x| \} \in R_2$$ which one is correct, given that vector $v = [x_1,y_1]$ and $u = [x_2,y_2]$ are in $W$? $$ 1.\ v+u = [x_1+x_2, \lVert x_1+x_2 \lVert\ ] $$ $$ 2.\ v+u = [x_1 , \lVert x_1 \lVert\ ] + [x_2 , \lVert x_2 \lVert\ ] = [x_1+x_2, \lVert x_1\lVert + \lVert x_2 \lVert\ ] $$ I don't think they are equal, given that if $x_1 = -1$ and $x_2 = 1$, equation $1$ would give $[0, 0]$ while equation $2$ would give $[0, 2]$.

edit: $$W = \{[x, y] \mid y = |x| \} \in R_2$$

mattyboi
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  • What do you mean by $y=|x|\in R_2$? Do you mean $x\in\mathbb{R}^2$ (in which case $y=||x||\in\mathbb{R}$)? Or do you mean $\mathbb{R}$ instead of $R_2$? – A. Goodier Dec 17 '17 at 18:55
  • And in case $R_2$ is indeed supposed to be $\mathbb R$ and $\lVert\ldots\rVert$ to be $|\ldots|$, one has $[x_1,y_1] + [x_2,y_2] = [x_1+x_2,y_1+y_2]$, and then substitutes $y_i = |x_i|$ and gets (2), not (1). – arseniiv Dec 17 '17 at 19:55
  • Please see edit. Thanks @arseniiv – mattyboi Dec 17 '17 at 20:29

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