Definition Let $V$ be a topological vector space over a subfield of the complex numbers (usually taken to be $\mathbb{R}$ or $\mathbb{C}$ ). If the topology of $V$ has a basis where each member is a convex set, then $V$ is a locally convex topological vector space [1].

Though most vector spaces occurring in practice are locally convex, the spaces $L^p$ for $0<p<1$ are examples of spaces which are not locally convex.

References

1

G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.