A Sobolev space is a mathematical concept used in functional analysis and partial differential equations. It consists of functions that have certain smoothness properties and whose derivatives also belong to a specific space of integrable functions. This allows for the study of functions that may not be differentiable in the traditional sense but still exhibit controlled behavior. Sobolev spaces are denoted as W^{k,p} where k indicates the number of derivatives considered, and p represents the integrability condition. These spaces are essential in various applications, including theory of weak solutions and variational methods, providing a framework for analyzing problems in mathematical physics and engineering.