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2 Basic properties of the spaces Lp
3 Linear functionals on Lp
4 Compactness of spaces
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absolutely continuous adjoint Analogously arbitrary function averaged functions boundary value bounded domain bounded function boundedness Cauchy problem characteristic cone choose coefficients consequently const constant construct continuous derivatives continuous functions continuously differentiable converges strongly coordinates corresponding defined definition denote derivatives of arbitrary derivatives of order Dirichlet problem du/dt element equal to zero equicontinuous equivalent estimate exists finite number follows formula func functional on Lp given Holder inequality homogeneous functions hyperplane Imbedding Theorem initial data integral linear functional manifold minimizing sequence Minkowski inequality norm obtain Obviously plane polynomial polynomial of degree Proof respect right hand side satisfying the conditions sequence of functions solution space Lp strongly compact sufficiently smooth summable function Suppose surface theorem is proved tion triangle inequality unit sphere vanish variables vector virtue wave equation weakly convergent whole space xux2