The text presents a mathematical proof showing that the spaces H and W are equivalent. H is defined as the closure in W of smooth functions in Ω intersected with W, while W is the space of functions with distribution derivatives up to order m in L^p(Ω). The theorem states that H is equivalent to W. The proof uses a lemma that shows that any function with distribution derivatives in Ω can be approximated by smooth functions in W with arbitrary precision. The proof involves constructing a sequence of smooth functions that approximate the given function and showing that the difference between the function and the sequence is in W with norm less than ε. The conclusion is that the distinction between weak and strong derivatives is no longer necessary, as H and W are the same. The text also notes that some authors define H as the closure of functions smooth up to the boundary of Ω, in which case H and W are not the same unless the boundary is smooth.The text presents a mathematical proof showing that the spaces H and W are equivalent. H is defined as the closure in W of smooth functions in Ω intersected with W, while W is the space of functions with distribution derivatives up to order m in L^p(Ω). The theorem states that H is equivalent to W. The proof uses a lemma that shows that any function with distribution derivatives in Ω can be approximated by smooth functions in W with arbitrary precision. The proof involves constructing a sequence of smooth functions that approximate the given function and showing that the difference between the function and the sequence is in W with norm less than ε. The conclusion is that the distinction between weak and strong derivatives is no longer necessary, as H and W are the same. The text also notes that some authors define H as the closure of functions smooth up to the boundary of Ω, in which case H and W are not the same unless the boundary is smooth.