Definition 11.1.10 (Topology Induced by Seminorm).label Let $E$ be a vector space over $K \in \RC$ and $\seqi{[\cdot]}$ be seminorms, then:

1. (1)

   For each $i \in I$, $d_{i}: E \times E \to [0, \infty)$ defined by $(x, y) \mapsto [x - y]_{i}$ is a pseudo-metric.

2. (2)

   The topology induced by $\seqi{d}$ makes $E$ a topological vector space.

3. (3)

   For each $i \in I$, $[\cdot]_{i}: E \to [0, \infty)$ is continuous.

The topology induced by $\seqi{d}$ is the vector space topology induced by $\seqi{[\cdot]}$. In addition,

1. (U)

   For any family $\seqj{[\cdot]}$ of seminorms continuous on $E$, the vector space topology induced by $\seqj{[\cdot]}$ is contained in the vector space topology induced by $\seqi{[\cdot]}$.