One reason why locally convex TVS are important is that lots of (continuous!) linear functionals exist on them, at least if one assumes an appropriate choice principle, e.g., axiom of choice or ultrafilter theorem (or just dependent choice for a separable space). This fact is encapsulated in the Hahn-Banach theorem; a nice exposition is given in Terry Tao’s lecture notes. By way of contrast, a TVS which is not locally convex, such as the topological vector space $L^p([0, 1])$ where $0 \lt p \lt 1$, need not have any (nonzero) functionals at all.

The collections of functionals on a LCTVS is used in a way analogous to the collection of coordinate projections $pr_i:\mathbb{R}^n\to \mathbb{R}$. For example, curves in a LCTVS over the reals can be composed with functionals to arrive at a collection of functions $\mathbb{R} \to \mathbb{R}$ which are analogous to the ‘components’ of the curve.

In one respect, a locally convex TVS is a nice topological space in that there are enough co-probes by maps to the base field.

Diagram of properties

References

J. L. Taylor, Notes on locally convex topological vector spaces (1995) (pdf)