Locally Convex Topological Vector Spaces (LCTVS)

Locally Convex Topological Vector Spaces (LCTVS)

On the Topological Vector Spaces (TVS) page we said that a linear space $X$ with a Hausdorff topology is said to be a topological vector space if $+$ and $\cdot$ are both continuous with respect to the topology.

On the Convex Subsets of Vector Spaces page that a subset $K$ of a vector space $X$ is said to be convex if for every pair of points $x, y \in K$ and for every $0 \leq t \leq 1$ we have that:

\begin{align} \quad tx + (1 - t)y \in K \end{align}

Furthermore, we said that elements of the form above are called convex combinations of $x$ and $y$.

We are now ready to define a locally convex topological vector space.

Definition: A normed linear space $X$ over $\mathbb{R}$ (or $\mathbb{C}$) is said to be a Locally Convex Topological Vector Space (abbreviated LCTVS) if $X$ is also equipped with a Hausdorff topology $\tau$ and such that:
1) The operation of addition $+ : X \times X \to X$ defined by $(x, y) \to x + y$ is continuous on $X \times X$.
2) The operator of scalar multiplication $\cdot : \mathbb{R} \times X \to X$ defined by $(\lambda, x) \to \lambda x$ is continuous on $\mathbb{R} \times X$.
3) There is a local base at the origin consisting of only convex sets.

Recall that in a topological vector space, a local base at the origin completely determines the topology on $X$ since we can translate a local base at $0$ to any point $x \in X$.

The simplest examples of locally convex topological vector spaces are normed linear spaces with the norm topology. We noted that these spaces are already topological vector spaces and we have already seen that open balls are convex sets.

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