One of the most useful tools in the local spectral theory of bounded or unbounded closed operators on Banach spaces is called the single-valued extension property (SVEP) (see [1, 2]). This notion was firstly introduced by Dunford [3] and studied by a variety of mathematicians. We can cite Finch [4], Laursen and Neumann [2] and Aiena and her co-authors [5–8]. We refer the reader to Finch’s article [4] for the definition of the single-valued extension property of the closed linear operator U on Banach space. Motivated by Aiena et al.’s work [9], we introduce some concepts of the local spectral theory and the single-valued extension property abbreviated SVEP, of the closed linear relations on a Banach space. After that, we analyze basic properties of these notions and establish a relationship between the analytic spectral subspace and the analytic core.