Vector spaces A vector space is a space that satisfies two requirements : 1. $\vec{v}+\vec{w}$ and $c\vec{v}$ are in the space. 2. all combinations $c\vec{v} + d\vec{w}$ are in the space. So, a real vector space is a set of vectors together with rules for vector addition and multiplication by real numbers. Examples of three spaces 1. The inifinite-dimensional space $\mathbb{R}^\infty$ is a space..