# Why do we study specifically 'normed' vector spaces?

When we study vector spaces, it is useful to define a norm on it for countless reasons.

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.