In the textbook 'Linear Algebra Done Right' by 'Sheldon Axler', the requirements for a subspace are listed as
- contains the additive identity
- is closed under addition
- is closed under scalar multiplication
There is no requirement for containing additive inverses.
So if subspaces are vector spaces themselves, shouldn't containing additive inverses also be a requirement??
Best Answer
There's no need for a separate axiom for the existence of additive inverses, since closure under scalar multiplication means $-1 \cdot v$ for any $v \in V$ and $-1 \cdot v$ is an additive inverse of $v$: $$-1 \cdot v + v = -1 \cdot v + 1 \cdot v = (-1 + 1) \cdot v = 0 \cdot v = 0.$$