I just read the definition of the quotient norm, but it had no examples, so I constructed my own. Is this correct?
Let $(X=\Bbb R^3,\|\cdot\|)$ with Euclidean norm and $Y\subset X$, $\quad Y=\{(a,0,0):a\in \Bbb R\}$
So $X/Y = \{ [(a,b,c)]=(a,b,c)+Y: a,b,c\in \Bbb R\}$
And we look at: $$\|[(3,6,4)]\|_{X/Y}= \|(3,6,4)-(a,0,0)\|_X$$ $$=\inf_{y\in Y}\left(\sqrt{(3-a)^2+6^2+4^2}\right)=\sqrt{(3-3)^2+6^2+4^2}=\sqrt{36+16}=\sqrt{52}=2\sqrt{13}$$
Is that the correct way to apply the quotient norm?