Theorem: Convex Hull of Union of Compact Set and Closed von Neumann-Bounded Set in Hausdorff Topological Vector Space is Closed

Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \tau}$ be a Hausdorff topological vector space.
Let $A \subseteq X$ be compact.
Let $B \subseteq X$ be closed and von Neumann-bounded.
Let $C = \map {\mathrm {conv} } {A \cup B}$ be the convex hull of $A \cup B$.
Then $C$ is closed.

Proof:
We show that every convergent net in $C$ has its limit in $C$.
Let $\struct {\AA, \preceq_{\AA} }$ be a directed set.
Let $\family {z_\alpha}_{\alpha \mathop \in \AA}$ be a net in $C$ converging to $z$.
We want to show that $z \in C$.
From Convex Hull of Finite Union of Convex Sets, there exists nets: 
:$\family {x_\alpha}_{\alpha \mathop \in \AA}$ in $A$
:$\family {y_\alpha}_{\alpha \mathop \in \AA}$ in $B$
:$\family {\lambda_\alpha}_{\alpha \mathop \in \AA}$ in $\closedint 0 1$
such that:
:$z_\alpha = \paren {1 - \lambda_\alpha} x_\alpha + \lambda_\alpha y_\alpha$ for each $\alpha \in \AA$.
Since $A$ is compact:
:there exists a directed set $\struct {\BB, \preceq_\BB}$ and a cofinal increasing mapping $\alpha : \BB \to \AA$ such that:
::$\struct {x_{\map \alpha \beta} }_{\beta \mathop \in \BB}$ is a subnet converging to $x \in A$
from Characterization of Compactness in terms of Nets.
Since $\closedint 0 1$ is compact from Closed Real Interval is Compact Space:
:there exists a directed set $\struct {\CC, \preceq_\CC}$ and cofinal increasing mapping $\beta : \CC \to \BB$ such that:
::$\struct {\lambda_{\map \alpha {\map \beta \gamma} } }_{\gamma \mathop \in \CC}$ is a subnet converging to $\lambda \in \closedint 0 1$
from Characterization of Compactness in terms of Nets.
From Subnet of Convergent Net Converges to Same Limit, $\struct {x_{\map \alpha {\map \beta \gamma} } }_{\gamma \mathop \in \CC}$ also converges to $x$.
For brevity, relabel $\struct {x_{\map \alpha {\map \beta \gamma} } }_{\gamma \mathop \in \CC}$, $\struct {y_{\map \alpha {\map \beta \gamma} } }_{\gamma \mathop \in \CC}$ and $\struct {\lambda_{\map \alpha {\map \beta \gamma} } }_{\gamma \mathop \in \CC}$ as simply $\struct {x_\gamma}_{\gamma \mathop \in \CC}$, $\struct {y_\gamma}_{\gamma \mathop \in \CC}$ and $\struct {\lambda_\gamma}_{\gamma \mathop \in \CC}$.
At this point we know that $x_\gamma \to x$, $y_\gamma \to y$ and $\lambda_\gamma \to \lambda$.
Suppose first that $\lambda > 0$.
Write:
:$\ds y_\gamma = \frac {z_\gamma - \paren {1 - \lambda_\gamma} x_\gamma} {\lambda_\gamma}$ for each $\gamma \in \CC$.
From Convergent Scalar Multiple of Convergent Net in Topological Vector Space is Convergent and Sum of Convergent Nets in Topological Vector Space is Convergent, we have:
:$\ds y_\gamma \to \frac {z - \paren {1 - \lambda} x} \lambda := y$
Since $B$ is closed, we have $y \in B$ from Characterization of Closedness in terms of Nets.
We can then write:
:$z = \paren {1 - \lambda} x + \lambda y$ 
Since $x \in A$ and $y \in B$, we have $x \in A \cup B$ and $y \in A \cup B$.
Hence $z \in \map {\mathrm {conv} } {A \cup B}$.
That is, $z \in C$ as desired.
Suppose now that $\lambda = 0$. 
From Convergent Scalar Multiple of Convergent Net in Topological Vector Space is Convergent and Sum of Convergent Nets in Topological Vector Space is Convergent, we have:
:$z_\gamma = \paren {1 - \lambda_\gamma} x_\gamma + \lambda_\gamma y_\gamma \to x$
From Characterization of Hausdorff Property in terms of Nets, $\family {z_\gamma}_{\gamma \mathop \in \CC}$ has a unique limit.
Hence $z = x$. 
So $z \in A$, and in particular $z \in \map {\mathrm {conv} } {A \cup B} = C$. 
{{qed}}
Category:von Neumann-Bounded Subsets of Topological Vector Spaces
Category:Convex Hulls
Category:Compact Topological Spaces
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