Representation of a linear functional in a Hilbert Sobolev space

In this paper, a representation of a linear functional in the Hilbert case of a Sobolev space is obtained. The space is normed by expression that is introduced for the first time and that has not been considered anywhere before. First, the intermediate problems of substantiating the legality of the norm, the inner product generating the norm, and the metrics generated with the norm are solved. As necessary, the proof of concomitant inequalities and identities is given. Second, a finite linear functional is considered. The fmiteness makes it possible to establish explicit structural elements included in the representation. Solution of the problem of finding a representing function belonging to the same space that a test function belongs to is based on the property of uniform convexity of a unit sphere using a limit element matching the functional. Such a way leads to a partial differential equation in generalized functions. The operator of the equation is linear and includes constant coefficients, which makes it possible to find a solution as a convolution of its fundamental solution and its right-hand side term. It is shown that the solution exists and is the required function representing the functional.

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