We prove that every evolution algebra is a normed algebra, for an -norm defined in terms of a fixed natural basis. We further show that a normed evolution algebra is a Banach algebra if and only if , where is finite-dimensional and is a zero-product algebra. In particular, every nondegenerate Banach evolution algebra must be finite-dimensional and the completion of a normed evolution algebra is therefore not, in general, an evolution algebra. We establish a sufficient condition for continuity of the evolution operator of with respect to a natural basis , and we show that need not be continuous. Moreover, if is finite-dimensional and , then is given by , where and is the multiplication operator , for . We establish necessary and sufficient conditions for convergence of , for all , in terms of the multiplicative spectrum of . Namely, converges, for all , if and only if and , where denotes the index of in the spectrum of .
"Analytic aspects of evolution algebras." Banach J. Math. Anal. 13 (1) 113 - 132, January 2019. https://doi.org/10.1215/17358787-2018-0018