Topology generated by a family of seminorms

Remarks

With this topology $V$ is a topological vector space, each $p \in \mathcal{P}$ is continuous, and it is the coarsest topology on $V$ with these two properties. A net $(x_i)$ converges to $x$ if and only if $p(x_i - x) \to 0$ for every $p \in \mathcal{P}$. The topology is Hausdorff if and only if $\mathcal{P}$ separates points: for each $x \ne 0$ some $p \in \mathcal{P}$ has $p(x) \ne 0$. Example: on $\R^X$, the space of all real functions on a set $X$, the seminorms $p_x(f) = \abs{f(x)}$ for $x \in X$ generate the topology of pointwise convergence, which is the product topology. The check: a basic set $B(f; p_{x_1}, \ldots, p_{x_n}; \varepsilon)$ restricts the values at $x_1, \ldots, x_n$ only, as a basic open set of the product topology does. Non-example: for $0 < p < 1$, the usual metric topology of $L^p([0, 1])$ is not generated by any family of seminorms. Indeed, its only continuous linear form is $0$. A nonzero continuous seminorm would give a nonzero continuous linear form by the Hahn–Banach theorem. The topologies generated by families of seminorms are exactly the topologies of locally convex spaces.