A note on vertex algebras and Costello-Gwilliam factorization algebras

We show that the construction of vertex algebras from Costello-Gwilliam factorization algebras on $\mathbb{C}$ can be achieved without the discreteness condition on the weight spaces. Furthermore, we construct locally constant factorization algebras from commutative vertex algebras, and discuss the relationship between this construction and the jet algebras of commutative algebras.