Example 13.1.20 (Infinite Dimensional Vector Space).label $V\define C(\R;\R)$ is a vector space over $\R$ under pointwise addition and scalar multiplication. We observe $\dim(V)=\infty$ since $\curl{1,e^x,...,e^{mx}}$ is a family of independent vectors such that $e^{kx}$ for $k>m\in\N$ is not in the span of this set.

We note that Corollary 13.1.19 is not true if $A$ is compact in an infinite dimensional vector space. This is because $Conv(A)$ may not be closed. ^[1]