The article by Norman G. Meyers and James Serrin, published in 1964, discusses the mathematical concepts of \( H \) spaces and \( W \) spaces, which are important in the theory of partial differential equations and the calculus of variations. The authors define these spaces as follows: - \( W^{m,p}(\Omega) \) is the Banach space of all complex-valued functions \( u = u(x) \) on an open set \( \Omega \) in \( R^n \) with distribution derivatives up to order \( m \) in \( L^p(\Omega) \), equipped with the norm \( \|u\|_{W^{m,p}(\Omega)} = \sum_{|\alpha| \leq m} \|D^\alpha u\|_{L^p(\Omega)} \). - \( H^{m,p}(\Omega) \) is the closure in \( W^{m,p}(\Omega) \) of the space \( C^\infty(\Omega) \cap W^{m,p}(\Omega) \). The key result presented in the article is that \( H \subset W \) and, more importantly, that \( H \equiv W \). This means that for any function \( u \) with distribution derivatives up to order \( m \) in \( L^p \) on \( \Omega \), there exists a function \( v \) in \( C^\infty(\Omega) \) such that \( u - v \) is in \( W^{m,p}(\Omega) \) and the norm difference is arbitrarily small. The proof involves using a partition of unity and a mollifier to approximate \( u \) by a smooth function \( v \). The authors also note that the distinction between weak and strong derivatives is no longer necessary since \( H \equiv W \). They recommend retaining the term "strong derivative" to avoid confusion. Finally, they point out that if the \( H \) spaces are defined by taking the closure of functions that are smooth up to the boundary of \( \Omega \), then \( H \) and \( W \) are not the same unless certain smoothness conditions are imposed on the boundary.The article by Norman G. Meyers and James Serrin, published in 1964, discusses the mathematical concepts of \( H \) spaces and \( W \) spaces, which are important in the theory of partial differential equations and the calculus of variations. The authors define these spaces as follows: - \( W^{m,p}(\Omega) \) is the Banach space of all complex-valued functions \( u = u(x) \) on an open set \( \Omega \) in \( R^n \) with distribution derivatives up to order \( m \) in \( L^p(\Omega) \), equipped with the norm \( \|u\|_{W^{m,p}(\Omega)} = \sum_{|\alpha| \leq m} \|D^\alpha u\|_{L^p(\Omega)} \). - \( H^{m,p}(\Omega) \) is the closure in \( W^{m,p}(\Omega) \) of the space \( C^\infty(\Omega) \cap W^{m,p}(\Omega) \). The key result presented in the article is that \( H \subset W \) and, more importantly, that \( H \equiv W \). This means that for any function \( u \) with distribution derivatives up to order \( m \) in \( L^p \) on \( \Omega \), there exists a function \( v \) in \( C^\infty(\Omega) \) such that \( u - v \) is in \( W^{m,p}(\Omega) \) and the norm difference is arbitrarily small. The proof involves using a partition of unity and a mollifier to approximate \( u \) by a smooth function \( v \). The authors also note that the distinction between weak and strong derivatives is no longer necessary since \( H \equiv W \). They recommend retaining the term "strong derivative" to avoid confusion. Finally, they point out that if the \( H \) spaces are defined by taking the closure of functions that are smooth up to the boundary of \( \Omega \), then \( H \) and \( W \) are not the same unless certain smoothness conditions are imposed on the boundary.