Rigged Hilbert Space of Free Coherent States and $p$-Adic Numbers

We investigate the rigged Hilbert space of free coherent states. We prove that this rigged Hilbert space is isomorphic to the space of generalized functions over a $p$-adic disk. We discuss the relation of the described isomorphism of rigged Hilbert spaces and noncommutative geometry and show that the considered example realizes the isomorphism between the noncommutative line and the $p$-adic disk.