Definition 1

— A topological vector space is locally convex (real) if there exists a fundamental system of neighbourhoods of 0 that are convex sets.

Such a space is called a locally convex space. Its topology is called a locally convex topology.

The topological vector spaces over $\mathbf{R}$ which we study in the rest of this book are nearly all locally convex.

If $V$ is a convex neighbourhood of 0 in the locally convex space $E$, then $V \cap (-V)$ is a symmetric convex neighbourhood of 0. As the closure of a convex set is convex (II, p. 13, prop. 14) it follows from I, p. 7, prop. 4 that the neighbourhoods of 0 in $E$ which are closed, symmetric and convex, form a fundamental system of neighbourhoods invariant under homotheties of centre 0 and ratio $\neq 0$.