The Gelfand-Naimark theorem says that any $C^\ast$-algebra is isomorphic to a norm closed $\ast$-subalgebra of $B(H)$ for a suitable Hilbert space $H$. In order to prove this theorem, it is imperative that our $C^\ast$-algebra have states, which uses the Hahn-Banach theorem.

One can also use the Hahn-Banach theorem instead of Tychonoff's theorem to construct $\beta X$, the Stone-Cech compactification of $X$. This is closely related to @Mark Meckes' answer.

The Gelfand-Naimark theorem says that any $C^\ast$-algebra is isomorphic to a norm closed $\ast$-subalgebra of $B(H)$ for a suitable Hilbert space $H$. In order to prove this theorem, it is imperative that our $C^\ast$-algebra have states, which uses the Hahn-Banach theorem.

One can also use the Hahn-Banach theorem instead of Tychonoff's theorem to construct $\beta X$, the Stone-Cech compactification of $X$. This is closely related to @Mark Meckes' answer.