Definition of quotient space: equivalence classes vs affine subsets

Question

Suppose $V$ is a vector space over any field and $U$ its subspace. Consider two different definitions of the quotient space $V/U$:

1. $V/U = \{v + U \mid v \in V\}$, i.e. the set of all affine subsets of $V$ parallel to $U$. This definition is used in Axler's Linear Algebra Done Right 3rd edition section 3.E.
2. Define an equivalence relation $\sim$ on $V$: $v \sim v' \text{ iff } v - v' \in U$. Define $V/U$ to be the set of all equivalence classes generated by $\sim$. This definition is used in Hackbusch's Tensor Spaces and Numerical Tensor Calculus 3.1.3.

What advantages and disadvantages do these two definitions have compared to each other

Answer 1

They are the same.

Whenever $v'\sim v$, then $v'=v+v'-v\in v+U$. Conversely, if $v'\in v+U$, then $v'-v\in v-v+U=U$, hence $v'\sim v$.