A topological vector space is locally convex if it has a base of its topology consisting of convex open subsets. Equivalently, it is a vector space equipped with a gauge consisting of seminorms. As with other topological vector spaces, a locally convex space (LCS or LCTVS) is often assumed to be Hausdorff.
Locally convex (topological vector) spaces are the standard setup for much of contemporary functional analysis.
A natural notion of smooth map between lctvs is given by Michal-Bastiani smooth maps.
One reason why locally convex topological vector spaces 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 Tao 09. By way of contrast, a topological vector space which is not locally convex, such as the Lebesgue space $L^p([0, 1])$ where $0 \lt p \lt 1$, need not have any (nonzero) continuous linear functionals at all.
(directed system of seminorms)
A family $\{p_q\}_{q \in Q}$ of seminorms on some real vector space $V$ is called directed if
(e.g. Infusino 17 def. 4.2.15)
(maximum envelope of seminorms)
Let $V$ be a real vector space equipped with a set $\{p_q \colon V \to \mathbb{R}\}_{q \in Q}$ of seminorms.
Then the maxima of non-empty finite subsets $J \subset Q$ of these seminorms
are themselves seminorms, and the set of them
generate the same topology on $V$ as the original $\{p_q\}$ do. Moreover, the system of maximum seminorms evidently form a filtered system according to def. 1.
(e.g. Infusino 17 prop. 4.2.13, 4.2.14)
(characterization of continuity for linear functionals by norm-bounds)
Let $V$ be a real vector space and $\tau$ a topology on $V$ that makes it a locally convex topological vector space, induced from a set of seminorms $\{p_q \colon V \to \mathbb{R}\}_{q \in Q}$. Consider a linear function
(directed system of seminorms) If the system of seminorms $\{p_q\}_{q \in Q}$ is directed (def. 1) then $L$ is a continuous function with respect to $\tau$, hence is a continuous linear functional, precisely if it is $q$-continuous for some $q \in Q$:
(general system of seminorms) Together with prop. 1 this means that for $\{p_q\}_{q \in Q}$ any set of seminorms (not necessarily directed), then $L$ is continuous precisely if there exists a inhabited finite subset of seminorms such that $L$ is bounded with respect to the maximum over this set:
(e.g. Infusino 17, prop. 4.6.1, corollary 4.6.2), or remark 3-4 4. here: pdf
The collections of continuous linear functionals on a LCTVS is used in a way analogous to the collection of coordinate projections $pr_i:\mathbb{R}^n\to \mathbb{R}$ out of a Cartesian space. 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.
J. L. Taylor, Notes on locally convex topological vector spaces (1995) (pdf)
Maria Infusino, Topological vector spaces 2017