The paper by Jacques Simon provides a characterization of compact sets in the space \( L^p(0, T; B) \), where \( 1 < p < \infty \) and \( B \) is a Banach space. The main result, Theorem 1, characterizes compact sets in \( L^p(0, T; B) \) or \( C(0, T; B) \) (for \( p = \infty \)) through two criteria: a space criterion involving the integral of functions over intervals and a time criterion involving the uniform convergence of translations. The paper also discusses partial compactness, where compactness is required in \( L^p(0, T; B) \) for a set \( F \) that is bounded in \( L^q(0, T; B) \) with \( q > p \). Theorems and corollaries are provided to illustrate these criteria and their applications, including estimates for approximated solutions and their derivatives or integrals in various spaces. The paper also explores the optimality of these conditions and compares them with previous results.The paper by Jacques Simon provides a characterization of compact sets in the space \( L^p(0, T; B) \), where \( 1 < p < \infty \) and \( B \) is a Banach space. The main result, Theorem 1, characterizes compact sets in \( L^p(0, T; B) \) or \( C(0, T; B) \) (for \( p = \infty \)) through two criteria: a space criterion involving the integral of functions over intervals and a time criterion involving the uniform convergence of translations. The paper also discusses partial compactness, where compactness is required in \( L^p(0, T; B) \) for a set \( F \) that is bounded in \( L^q(0, T; B) \) with \( q > p \). Theorems and corollaries are provided to illustrate these criteria and their applications, including estimates for approximated solutions and their derivatives or integrals in various spaces. The paper also explores the optimality of these conditions and compares them with previous results.