Definition 11.1.1 (Topological Vector Space).label Let $E$ be a vector space over $K \in \bracs{\real, \complex}$ and $\topo \subset 2^{E}$ be a topology. If

1. (TVS1)

   The addition map $E \times E \to E$ with $(x, y) \mapsto x + y$ is continuous.

2. (TVS2)

   The scalar multiplication map $K \times E \to E$ with $(\lambda, x) \mapsto \lambda x$ is continuous.

then the pair $(E, \topo)$ is a topological vector space.