# $S$-Ultralinear algebra in the space of tempered distributions

Carfì, David (2001) $S$-Ultralinear algebra in the space of tempered distributions. Accademia Peloritana dei Pericolanti - Classe di Scienze FF.MM.NN., 78-79 (1). pp. 105-130.

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The aim of this paper is to introduce some new concepts that provide the space of tempered distributions with others "linear" structures in addition to its natural structure of linear vector space. These new concepts give for the first time a precise mathematical meaning to the concepts of continuum linear superposition used in physics, engineering and in particular to the concept of "basis" in the "Physical Hilbert space" of quantum mechanics. To this end, we must definte these new concepts and we must establish some their properties. The new concepts are: 1) nets of class $S$ 2) operator generated by a $S$-nets of tempered distributions; 3) nets of class $\mathcal{LS}$; 4) ultralinear combinations of a net of class $\mathcal{LS}$; 5) product of two nets of class $\mathcal{LS}$; 6) the $S$-ultralinear span of a net of class $\mathcal{LS}$; 7) $S$-system of ultra generators; 8) $S$-extended linear independence; 9) $S$-ultra-bases; 10) contravariant components with respect to an $S$-ultra basis. Mathematics Subject Classification (1991): 46F10, 46F99, 47A05, 47N50, 70A05, 70B05, 81P05, 81Q99. Key words: Linear operator, tempered distribution, basis, quantum system, state, linear superposition, subspace, generators, linear independence, contravariant components.