Among all those non-normable(even better if not even metrizable) topological vector spaces, which are the ones that are of special interest or very useful (in math or other subjects)?
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https://math.stackexchange.com/questions/1075222/is-there-such-thing-as-an-unnormed-vector-space#1075236 mentions a few examples. – Arthur Feb 16 '23 at 12:47
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Simple and important examples are the various spaces of test functions used to construct distributiona, like the Schwartz space and others. Similarly, the space of C^\infty functions on a manifold has an important topology, the Whitney topology, which does not arise from a norm. Once you understand these examples it is easy to construct many others. In topology one likes also the space $\mathbb R^{(\infty)}$. – Mariano Suárez-Álvarez Feb 17 '23 at 03:18