Continuous Functions form a Vector Space
# Statement
Let $X$ be a Topological Space, and let $V$ be a Topological Vector Space over Topological Field $F$. Then $C(X, V)$ forms a Vector Space over $F$.
# Proof
Recall Functions to a Vector Space form a Vector Space. Therefore, we only need to check $C(X, V)$ is a Vector Subspace. Let $c \in F$ and $f,g \in C(X, V)$. Then, because Composition of Continuous Functions is Continuous and $V,F$ is a Topological Vector Space and Topological Field respectively, we have $c * f + g \in C(X, V)$. Therefore $C(X, V)$ is a Vector Subspace of $V^{X}$, and thus a Vector Space in its own right. $\blacksquare$