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We prove that the a.e. convergence of a sequence of functions bounded in      H 1 ( R d ) {H^1}({{\mathbf {R}}^d})       to a function in      L 1 ( R d ) {L^1}({{\mathbf {R}}^d})       implies weak convergence.

On weak convergence in 𝐻¹(𝑅^{𝑑})