Linear Subspace of a Real Vector Space

In the setting of Separable Real Normed Spaces and Hahn–Banach: Notation, let $V$ be a real vector space with zero vector $0_{V}$, and let $W\subseteq V$.

(i) Zero. $0_{V}\in W$.

(ii) Sums. $u+v\in W$ for all $u,v\in W$.

(iii) Multiples. $\lambda u\in W$ for all $\lambda\in\mathbb{R}$ and $u\in W$.

(iv) Linear subspace. $W$ is a linear subspace of $V$ if it satisfies (i), (ii) and (iii).