The argument for the existence of discontinuous linear maps on normed spaces can be generalized to all metrisable topological vector spaces, especially to all Fréchet-spaces, but there exist infinite-dimensional locally convex topological vector spaces such that every functional is continuous.{{cn}}<ref>For example, the weak topology w.r.t. the space of all (algebraically) linear functionals.</ref> On the other hand, the [[Hahn–Banach theorem]], which applies to all locally convex spaces, guarantees the existence of many continuous linear functionals, and so a large dual space. In fact, to every convex set, the [[Minkowski gauge]] associates a continuous [[linear functional]]. The upshot is that spaces with fewer convex sets have fewer functionals, and in the worst-case scenario, a space may have no functionals at all other than the zero functional. This is the case for the [[Lp space|''L''<sup>''p''</sup>('''R''',''dx'')]] spaces with 0 < ''p'' < 1, from which it follows that these spaces are nonconvex. Note that here is indicated the [[Lebesgue measure]] on the real line. There are other ''L''<sup>''p''</sup> spaces with 0 < ''p'' < 1 which do have nontrivial dual spaces.
Another such example is the space of real-valued [[measurable function]]s on the unit interval with [[quasinorm]] given by
