In this expository article we present the characterizations proved by J. Chung, S.-Y. Chung and D. Kim, and by S.-Y. Chung, D. Kim and S. Lee, of the Schwartz space 𝔖 and of the Beurling-Björck Space 𝔖ω. For the most part we follow the original proofs, only adjusting the estimates and necessary in order to prove not only set-theoretic equalities, but also topological equalities. These results show that space 𝔖ω is, as a set as well as topologically, a direct generalization of the space 𝔖. Minor modifications of the original arguments allow us to obtain explicit linear estimates.