Definition 4.1.6 (Topology Induced by Seminorm).label Let $E$ be a vector space over $K\in \curl{\R,\com}$ and $\curl{\rho_i}_{i\in I}$ be a collection of seminorms then:

1. (1)

   For each $i\in I$, $d_{i}:E\ti E\to [0,\infty)$ defined by $(x,y)\mto \rho_{i}(x-y)$ is a pseudo-metric

2. (2)

   The topology induced by $\curl{d_i}_{i\in I}$ makes $E$ a topological vector space on which $\rho_{i}:E\to[0,\infty)$ is continuous for each $i\in I$

The topology induced by $\curl{d_i}_{i\in I}$ is the vector space topology induced by $\curl{\rho_i}_{i\in I}$. Observe

1. (1)

   For any family of $\curl{\rho_j}_{j\in J}$ of continuous seminorms on $E$, the vector space topology induced by $\curl{\rho_j}_{j\in J}$ is contained in the vector space topology induced by $\curl{\rho_i}_{i\in I}$