= Locally convex space
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A locally convex space is a <topological vector space> with a <neighbourhood basis> at zero consisting of <convex sets>. Equivalently, its <topology> is a <seminorm topology> generated by some family $\mathcal P$ of <seminorms>. We allow non-<Hausdorff> spaces here; a <separating family of seminorms> gives precisely the <Hausdorff> case. Every <normed vector space> is locally convex, but <smooth convergence> generally requires a whole family of measurements.
The geometric and measurement descriptions fit together: <seminorm> sublevel sets are <convex>, while the <Minkowski functional> of an open, convex zero <neighbourhood> invariant under all <scalar multiplications> with $|\lambda|\leq1$ is a <continuous> <seminorm>. Such <neighbourhoods> can be chosen as a <neighbourhood basis>. https://www.math.lmu.de/~philip/publications/lectureNotes/philipPeter_FunctionalAnalysis.pdf[Peter Philip, Functional Analysis, Theorem 1.40, equivalence of local convexity and seminorm topologies]{ref}.
Local convexity makes <Hahn-Banach theorem> and <separation of a point and an open convex set> available. It turns geometric distinctions into <continuous linear functionals>, connecting <convex sets> to the <continuous dual space>. For a <seminorm topology> generated by $\mathcal P$, a <linear functional> $\ell$ is <continuous> exactly when there are $p_1,\ldots,p_m\in\mathcal P$ and $C>0$ with
$$
|\ell(x)|\leq C\max_{1\leq i\leq m}p_i(x)\qquad(x\in E).
$$
Thus one <continuous linear functional> needs only finitely many of the available measurements. For example, <point evaluation> of a <derivative> on the <space of smooth functions on a compact interval> is controlled by a suitable <derivative seminorm>.
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