Definition 4.1.1 (Topological Vector Space).label Let $E$ be a vector space over $K\in \curl{\R,\com}$ and $\cali{T}\suf 2^{E}$ be a topology. We say that the pair $(E,\cali{T})$ is a topological vector space if the following holds:

1. (TV1)

   $E\ti E\to E$ defined by $(x,y)\mto x+y$ is continuous

2. (TV2)

   $K\ti E\to E$ defined by $(\lam,x)\mto \lam x$ is continuous