Given a vector space $V$ over a field $K$, let $\mathrm{End}(V)$ denote the algebra of linear endomorphisms of $V$. If $V$ is finite-dimensional, then it is well-known that the diagonalizable subalgebras of $\mathrm{End}(V)$ are characterized by their internal algebraic structure: they are the subalgebras isomorphic to $K^n$ for some natural number $n$. In case $V$ is infinite dimensional, the diagonalizable subalgebras of $\mathrm{End}(V)$ cannot be characterized purely by their internal algebraic structure: one can find diagonalizable and non-diagonalizable subalgebras that are isomorphic. I will explain how to characterize the diagonalizable subalgebras of $\mathrm{End}(V)$ as \emph{topological} algebras, using a natural topology inherited from $\mathrm{End}(V)$. I also hope to show how this characterization relates to an infinite-dimensional Wedderburn-Artin theorem that characterizes ``topologically semisimple'' algebras.