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.