Locally Convex Topological Vector Spaces

Locally Convex Topological Vector Spaces

Recall from 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:

(1)
\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 now define what it means for a vector space to be a locally convex topological vector space.

Definition: A Locally Convex Topological Vector Space (LCTVS) is a vector space $X$ with a Hausdorff topology with the following properties:
1) Vector addition is continuous, i.e., the map $+ : X \times X \to X$ with $(x, y) \to x + y$ is continuous.
2) Scalar multiplication is continuous, i.e., the map $\cdot : \mathbb{C} \times X \to X$ with $(\lambda, x) \to \lambda x$ is continuous.
3) There exists a local base at $0$ that consists a convex sets.
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