Sum of Convergent Nets in Topological Vector Space is Convergent

Theorem

Let $K$ be a topological field.

Let $X$ be a topological vector space over $K$.

Let $\struct {\Lambda, \preceq}$ be a directed set.

Let $x, y \in X$.

Let $\family {x_\lambda}_{\lambda \in \Lambda}$ and $\family {y_\lambda}_{\lambda \in \Lambda}$ be nets converging to $x$ and $y$ respectively.

Then the net $\family {x_\lambda + y_\lambda}_{\lambda \in \Lambda}$ converges to $x + y$.

Proof

For ease of reading, let $\succeq$ be the inverse relation of $\preceq$.

Let $W$ be an open neighborhood of $x + y$.

From Open Neighborhood of Point in Topological Vector Space contains Sum of Open Neighborhoods there exists an open neighborhood $U$ of $x$ and an open neighborhood $V$ of $y$ such that:

$U + V \subseteq W$

Since $\family {x_\lambda}_{\lambda \in \Lambda}$ converges to $x$, there exists $\lambda_1 \in \Lambda$ such that:

$x_\lambda \in U$ for $\lambda \succeq \lambda_1$.

Since $\family {y_\lambda}_{\lambda \in \Lambda}$ converges to $y$, there exists $\lambda_2 \in \Lambda$ such that:

$y_\lambda \in V$ for $\lambda \succeq \lambda_2$.

Since $\struct {\Lambda, \preceq}$ is directed, there exists $\lambda_\ast \in \Lambda$ such that:

$\lambda_\ast \succeq \lambda_1$ and $\lambda_\ast \succeq \lambda_2$.

Since $\preceq$ is transitive, if $\lambda \in \Lambda$ has $\lambda \succeq \lambda_\ast$, then $\lambda \succeq \lambda_1$ and $\lambda \succeq \lambda_2$.

Hence, for $\lambda \succeq \lambda_\ast$ we have $x_\lambda \in U$ and $y_\lambda \in V$.

Hence:

$x_\lambda + y_\lambda \in U + V \subseteq W$ for $\lambda \succeq \lambda_\ast$.

Since $W$ was an arbitrary open neighborhood of $x + y$, the net $\family {x_\lambda + y_\lambda}_{\lambda \in \Lambda}$ converges to $x + y$.

$\blacksquare$