An introduction to S Bases in S Linear Algebra

  • David CARFI’ University of California Riverside
DOI: https://doi.org/10.14505/jmef.v5.2(9).03

Abstract

In this lecture note we define the S bases for the spaces of tempered distributions.
These new bases are the analogous of Hilbert bases of separable Hilbert spaces for the continuous case (they are indexed by m-dimensional Euclidean spaces) and enjoy properties similar to those shown by algebraic bases in the finite dimensional case.
The S bases are one possible rigorous and extremely manageable mathematical model for the "physical" bases used in Quantum Mechanics.

References

[1]. I. Antoniou and I. Prigogine. Intrinsic irreversibility and integrability of dynamics. Physica, A192:443-464, 1993.
[2]. R. Balescu. Equilibrium and Nonequilibrium Statistical Mechanics. Wiley & Sons, New York, 1975.
[3]. J. Barros-Neto. An Introduction to the theory of distributions. Marcel Dekker, New York, 1973.
[4]. N. Boccara. Functional Analysis: An Introduction for Physicists. Academic Press, Boston, 1990.
[5]. N. Bourbaki. Topologie Générale. (Fascicule de Résultats). Hermann, Paris, 1953.
[6]. N. Bourbaki. Topological Vector Spaces. Hermann, Paris, 1955.
[7]. N. Bourbaki. Intégration. Chapitre 1 à 4. Hermann, Paris, 1965.
[8]. D. Carfì. S-Ultralinear Algebra in the space of tempered distributions. AAPP| Physical, Mathematical, and Natural Sciences, 78-79(1):105-130, 2001. http://cab.unime.it/mus/664/
[9]. D. Carfì. SL-ultradifferentiable Manifolds. Analele Universitatii Bucuresti - Seria Informatica, 50:21-31, 2001. Proceedings of the Centellian of Vranceanu.
[10]. D. Carfì. Dirac-orthogonality in the space of tempered distributions. Journal of Computational and Applied Mathematics, 153(1-2):99-107, 2003. 6th International Symposium on Orthogonal Polynomials, Special Functions and Applications. Elsevier. http://dx.doi.org/10.1016/S0377-0427(02)00634-9
[11]. D. Carfì. S-ultralinear operators in Quantum Mechanics. In M. Moreau, E. Hideg, K. Martinàs, and D. Meyer, editors, Complex systems in natural and social sciences. (Proceedings of the 7th Workshop on Complex Systems in Natural and Social Sciences, Màtrafüred, Hungary, September 26-29, 2002), pages 33-46. ELFT, Budapest, 2003.
[12]. D. Carfì. S-linear operators in quantum Mechanics and in Economics. Applied Sciences (APPS), 6(1):7-20, 2004. http://www.mathem.pub.ro/apps/v06/A06.htm
[13]. D. Carfì. Tangent spaces on S-manifolds. Differential Geometry Dynamical Systems, 6:1-13, 2004. http://www.mathem.pub.ro/dgds/v06/D06-CAR3.pdf
[14]. D. Carfì. S-diagonalizable operators in Quantum Mechanics. Glasnik Mathematicki, 40(2):261-301, 2005. http://dx.doi.org/10.3336/gm.40.2.08
[15]. D. Carfì. Quantum statistical systems with a continuous range of states. In M. Primicerio, R. Spigler, and V. Valente, editors, Applied and Industrial Mathematics in Italy (Proceedings of the 7th Conference, Venice, Italy, 20 - 24 September 2004), volume 69 of Series on Advances in Mathematics for Applied Sciences, pages 189{200. World Scientific, 2005. http://dx.doi.org/10.1142/9789812701817_0018 .
[16]. D. Carfì. An S-Linear State Preference Model. In Communications to SIMAI, volume 1, pages 1-4, 2006. https://dx.doi.org/10.1685/CSC06037 . Also available, in extended version, as Researchgate Paper at https://dx.doi.org/10.13140/RG.2.1.1006.2800
[17]. D. Carfì. Dyson formulas for Financial and Physical evolutions in S’n. Communications to SIMAI Congress, 2:1-10, 2007. https://dx.doi.org/10.1685/CSC06156
[18]. D. Carfì. Feynman's transition amplitudes in the space S’n. AAPP | Physical, Mathematical, and Natural Sciences, 85(1):1-10, 2007. http://dx.doi.org/10.1478/C1A0701007
[19]. D. Carfì. S-Linear Algebra in Economics and Physics. Applied Sciences (APPS), 9:48-66, 2007. http://www.mathem.pub.ro/apps/v09/A09-CA.pdf
[20]. D. Carfì. Topological characterizations of S-linearity. AAPP | Physical, Mathematical, and Natural Sciences, 85(2):1-16, 2007. http://dx.doi.org/10.1478/C1A0702005
[21]. D. Carfì. Superpositions in Prigogine's approach to irreversibility for physical and financial applications. AAPP | Physical, Mathematical, and Natural Sciences, 86(S1):1-13, 2008. https://dx.doi.org/10.1478/C1S0801005
[22]. D. Carfì. Foundations of Superposition Theory, volume 1. Il Gabbiano, 2010. ISBN: 978-88-96293-11-9. https://dx.doi.org/10.13140/RG.2.1.3352.2642
[23]. D. Carfì. The pointwise Hellmann-Feynman theorem. AAPP | Physical, Mathematical, and Natural Sciences, 88(1):1-14, 2010. http://dx.doi.org/10.1478/C1A1001004
[24]. D. Carfì. S-Bases in S-Linear Algebra. ArXiv Paper, pages 1-11, 2011. http://arxiv.org/abs/1104.3324
[25]. D. Carfì. Multiplicative operators in the spaces of Schwartz families. ArXiv Paper, pages 1{15, 2011. http://arxiv.org/abs/1104.3908
[26]. D. Carfì. Schwartz families in tempered distribution spaces. ArXiv Paper, pages 1-15, 2011. http://arxiv.org/abs/1104.4651
[27]. D. Carfì. Schwartz Linear operators in distribution spaces. ArXiv Paper, pages 1-14, 2011. http://arxiv.org/abs/1104.3380
[28]. D. Carfì. Spectral expansion of Schwartz linear operators. ArXiv Paper, pages 1-23, 2011. http://arxiv.org/abs/1104.3647
[29]. D. Carfì. Summable families in tempered distribution spaces. ArXiv Paper, pages 1-7, 2011. http://arxiv.org/abs/1104.4660
[30]. D. Carfì. Motivations and origins of Schwartz Linear Algebra in Quantum Mechanics. Researchgate Paper, pages 1-6, 2014. https://dx.doi.org/10.13140/2.1.1447.1361
[31]. D. Carfì. Quantum Mechanics and Dirac Calculus in Schwartz Distribution Spaces, vol. 1. Superpositions in Distribution Spaces, Postulates of Quantum Mechanics in S’n, Schwartz Linear Algebra, S-Representation in Quantum Mechanics, Dirac Orthogonality, S-Linear Quantum Statistics. Il Gabbiano, 2014. https://dx.doi.org/10.13140/2.1.4959.1360
[32]. D. Carfì. Spectral expansion of Schwartz linear operators. Researchgate Paper, pages 1-23, 2015. https://dx.doi.org/10.13140/RG.2.1.3688.7762
[33]. D. Carfì. Differential Geometry and Relativity Theories: tangent vectors, derivatives, paths, 1-forms. Journal of Mathematical Economics and Finance, 2(1(2)):85-127, 2016. http://journals.aserspublishing.eu/jmef/article/view/590
[34]. D. Carfì. Differential Geometry and Relativity Theories. Vol.1. Tangent vectors, tangent maps, paths, 1-forms. Il Gabbiano, 2016. ISBN: 978-88-96293-22-5.
[35]. D. Carfì. Motivations and origins of Schwartz Linear Algebra in Quantum Mechanics. Journal of Mathematical Economics and Finance, 2(2(3)):67-76, 2016. https://journals.aserspublishing.eu/jmef/article/view/923
[36]. D. Carfì. Differential Geometry and Relativity Theories vol 1. - Tangent vectors, derivatives, paths, 1-forms, vector fields. Lambert Academic Publishing, 2017. ISBN: 978-3-330-02885-2.
[37]. D. Carfì. Position operator. Journal of Mathematical Economics and Finance, 4(1(6)):79-87, 2018. https://journals.aserspublishing.eu/jmef/article/view/2404
[38]. D. Carfì, A. Caterino, and R. Ceppitelli. State preference models and jointly continuous utilities. In APLIMAT 2016 - 15th Conference on Applied Mathematics 2016, Proceedings, volume 1, pages 163-176. Slovak University of Technology in Bratislava, 2016. http://www.proceedings.com/29878.html
[39]. D. Carfì and C. Germanà. Some properties of a new product in S’n. Journal of Computational and Applied Mathematics, 153(1-2):109-118, 2003. 6th International Symposium on Orthogonal Polynomials, Special Functions and Applications. Elsevier. http://dx.doi.org/10.1016/S0377-0427(02)00635-0
[40]. D. Carfì and M. Magaudda. Superpositions in Distributions spaces. AAPP | Physical, Mathematical, and Natural Sciences, 85(2):1-14, 2007. http://dx.doi.org/10.1478/C1A0702006
[41]. D. Carfì and G. Orlando. Transposable Schwartz families. Researchgate Paper, pages 1-15, 2015. https://dx.doi.org/10.13140/RG.2.1.4561.8643
[42]. J.A. Dieudonné. La dualité dans les espaces vectoriels topologiques. Annales scietifiques de l' Ecole Normale Supérieure 3e série, 59:107-139, 1942.
[43]. J.A. Dieudonné and L. Schwartz. La dualité dans les espaces (F) and (LF). Annales de l'Institut Fourier, 1:61-101, 1949.
[44]. P.A.M. Dirac. The Principles of Quantum Mechanics. Oxford, the Clarendon Press, 1930.
[45]. J. Horvàth. Topological Vector Spaces and Distributions, volume 1. Addison- Wesley Publishing Company, 1966.
[46]. S. Kesavan. Topics in Functional Analysis and Applications. Wiley, New Delhi, 1989.
[47]. S. Lang. Real and functional analysis. Springer Verlag, 1993.
[48]. C. Miranda. Istituzioni di Analisi Funzionale. Unione Matematica Italiana, 1978.
[49]. W. Pauli. Die allgemeinen Prinzipien der Wellenmechanik. Springer Verlag, 1958.
[50]. R. Penrose. Quantum Mechanics: Foundations. Encyclopedia of Mathematical Physics, pages 260-265, 2006.
[51]. I. Prigogine. Non-Equilibrium Statistical Mechanics. Wiley, New York, 1962.
[52]. I. Prigogine. From Being to Becoming: Time and Complexity in the Physical Sciences. Freeman, San Francisco, 1980.
[53]. I. Prigogine. Le leggi del chaos. Laterza, Roma-Bari, 1993.
[54]. L. Schwartz. Functional Analysis. New York University, Courant Institute of Mathematical Sciences, 1964.
[55]. L. Schwartz. Mathematics for the Physical Sciences. Hermann and Addison-Wesley, 1966.
[56]. L. Schwartz. Théorie des Distributions. Hermann, Paris, 1966.
[57]. L. Schwartz. Application of distributions to the theory of elementary particles in quantum mechanics. Gordon and Breach, New York, 1968.
[58]. L. Schwartz. Analyse Hilbertienne. Hermann, Paris, 1979.
[59]. L. Schwartz. Oeuvres Scientifiques I, II, III. American Mathematical Society, 2011.
[60]. R. Shankar. Principles of Quantum Mechanics. Plenum Press, New York, 1994.
[61]. P. Shields. The Theory of Bernoulli Shifts. In Lectures in Mathematics. University of Chicago Press, Chicago, 1973.
[62]. F. Tréves. Topological Vector Spaces, Distributions and Kernels. Dover Books on Mathematics. Dover Publications, 2006.
[63]. K. Yosida. Functional Analysis (6th ed.). Classics in Mathematics. Springer, 1996.
[64]. E. Zeidler. Applied Functional Analysis, volume 1. Springer Verlag, 1995.
Published
2019-12-31
How to Cite
CARFI’, David. An introduction to S Bases in S Linear Algebra. Journal of Mathematical Economics and Finance, [S.l.], v. 5, n. 2, p. 35 - 49, dec. 2019. ISSN 2458-0813. Available at: <https://journals.aserspublishing.eu/jmef/article/view/4786>. Date accessed: 22 dec. 2024. doi: https://doi.org/10.14505/jmef.v5.2(9).03.
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Vol 5 No 2 (2019): JMEF, Volume V, Issue 2 Winter 2019
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