TITLE: Why isn't the quotient space $V/V = \{ V \}$?
QUESTION [11 upvotes]: If $W \subset V$, then one defines the quotient space, $$V/W = \{ v + W : v \in V \}$$
So why isn't this right? 
$$V/V = \{v + V : v \in V \} = \{V \}$$?
I read that $V/V = \{ 0 \}$? Why can't the whole set $V/V$ be partition, by $V$ itself?

REPLY [5 votes]: You are right, $V/V=\{V\}$. The reason this is sometimes written $\{0\}$ is that $V$ functions as the zero vector in $V/V$, i.e., in $V/V$, $V=0$.