Sobolev spaces are mathematical constructs used in functional analysis and partial differential equations. They provide a framework for studying functions that have certain smoothness properties and integrability. Specifically, Sobolev spaces allow for the inclusion of functions whose derivatives may not be continuous but are still square-integrable, making them useful in various applications. These spaces are denoted as W^{k,p} where k indicates the order of derivatives and p represents the integrability condition. Sobolev spaces play a crucial role in modern analysis, particularly in the study of weak solutions to differential equations, enabling mathematicians to work with more generalized function spaces.