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When we study vector spaces, it is useful to define a norm on it for countless reasons.
I was thinking about this recently and realised
Don't all vector spaces have norms on them? If they all have norms defined on them, why do we need to say specifically "normed vector space"?
If it is indeed the case that not all vector spaces have norms defined on them, then what is a simple example of a not-normed vector space, for a lack of a better term?
Normed spaces are sets along with norms. If you want to remove the norm, and just treat it as a space, you're free to do so. We say "normed" if we have a norm in mind. Some spaces are "normable" but we haven't chosen a specific norm. And different norms induce different topologies, in particular different ways of things converging. As you're possibly aware, this can greatly affect the overall properties of a space, such as convergence of sequences in the space or not. In the case of $L^2$ as a norm, that special case gets a free isometry with its dual, so the choice of norm can change things you observe.
Any non locally-convex space is even more without a norm, it is not even normable, since the topology of a norm is always locally convex. Take $\Bbb R$ with the $L^{p}$ "norm" with $p<1$ for example (it's not really a norm, but that's the point). An explicit example is, $\Bbb R^2$ with the topology generated by $\lVert (x,y)\rVert = (\sqrt{|x|}+\sqrt{|y|})^2$. This fails the triangle inequality so it is not a norm.
