Beyond Normed Spaces
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. 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(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 Lp spaces with 0 < p < 1 which do have nontrivial dual spaces.
Another such example is the space of real-valued measurable functions on the unit interval with quasinorm given by
This non-locally convex space has a trivial dual space.
One can consider even more general spaces. For example, the existence of a homomorphism between complete separable metric groups can also be shown nonconstructively.
Read more about this topic: Discontinuous Linear Map
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