Locally convex space

Remarks

Equivalently, the topology of $E$ is the topology generated by a family of seminorms, namely the family of all continuous seminorms on $E$. Many authors also require $E$ to be Hausdorff. Example: every normed space is locally convex, since the open balls $\set{x : \norm{x} < r}$ are convex and form a neighbourhood base at $0$. Convexity of a ball: if $\norm{x}, \norm{y} < r$ and $0 \le t \le 1$, then $\norm{t x + (1 - t) y} \le t \norm{x} + (1 - t) \norm{y} < r$. Non-example: for $0 < p < 1$, the space $L^p([0, 1])$ with the metric $d(f, g) = \int_0^1 \abs{f - g}^p$ is a topological vector space. But its only convex open set containing $0$ is the whole space, so it is not locally convex.