Let $(X,\tau)$ be a topological linear space. Is it true that the following equivalence hold?
$(X,\tau)$ is a locally convex space $\Longleftrightarrow$ There is a familly of seminorms $\mathcal{P}=(p_i)_{i\in I}$ such that $\tau$ is exactly the initial topology generated on $X$ by the familly $p_i:X\to (\mathbb{R},\tau_0),\ i\in I$.
By initial topology I mean the coarsest topology on $X$ for which the maps $p_i:X\to \mathbb{R}$ are continuous.
In all text that I've found there is an ambiguity about that, because the local convex spaces are introduced via a given local base of the origin wrt a family of seminorms.
P.S. By $\tau_0$ I mean the usual topology on $\mathbb{R}$.
