# necessary condition for subspace of a vector space

Currently I'm reading linear algebra books of leon's and friedberg's. In friedberg's book, for being subspace, a subset of vector space should (1). contain zero vector (2). closed under scalar multiplication (3). closed under vector addition

But condition (1) is missing in leon'book.

I think (1) is not necessary since if (2),(3) holds, then (1) must true

Does (1) necessary?

Probably it depends on your definition of vector space (i.e.: do you consider $\emptyset$ to be a vector space?) In my opinion, $\emptyset$ is should not be considered a vector space for various reasons, e.g. the fact that $\operatorname{span}\emptyset=\{0\}$, and thus point (1) should be included in the axioms for vector subspaces.