Bernstein, Non-standard analysis, Studies in Model Theory (M. Barwise (Editor), The handbook of mathematical logic, North Holland Studies in Logic, North-Holland, Amsterdam (in preparation).īastiani, Applications différentiables et variétés différentiables de dimension infinie, J. 374, Springer-Verlag, Berlin-New York., 1974. Sadayuki Yamamuro, Differential calculus in topological linear spaces, Lecture Notes in Mathematics, Vol.Luxemburg, Introduction to the theory of infinitesimals, Academic Press, New York-London, 1976. Abraham Robinson, Non-standard analysis, North-Holland Publishing Co., Amsterdam, 1966.94, Springer-Verlag, Berlin-New York, 1969. Moshé Machover and Joram Hirschfeld, Lectures on non-standard analysis, Lecture Notes in Mathematics, Vol.Papers in the foundations of mathematics. Luxemburg, What is nonstandard analysis?, Amer. Luxemburg (ed.), Applications of model theory to algebra, analysis, and probability., Holt, Rinehart and Winston, New York-Montreal, Que.-London, 1969. Knight, Solutions of differential equations in $B$-spaces, Duke Math. 417, Springer-Verlag, Berlin-New York, 1974. Hans Heinrich Keller, Differential calculus in locally convex spaces, Lecture Notes in Mathematics, Vol.Keisler, Elementary calculus: an approach using infinitesimals, Prindle, Weber and Schmidt, Boston, Mass. Moore Jr., The nonstandard theory of topological vector spaces, Trans. Godunov, The Peano theorem in Banach spaces, Funkcional. Dubinsky, Differential equations and differential calculus in Montel spaces, Trans. Bernstein, Non-standard analysis, Studies in model theory, Math. Andrée Bastiani, Applications différentiables et variétés différentiables de dimension infinie, J.Handbook of mathematical logic, Studies in Logic and the Foundations of Mathematics, vol. 90, North-Holland Publishing Co., Amsterdam, 1977.The technical improvement of our approach should lead to advances in a variety of subjects. It is simpler than many recent developments, e.g., Yamamuro and Keller. ![]() Though nontopologizable, our theory is a natural generalization of standard infinitesimal calculus (finite dimensional or Banach space), see Robinson, Keisler, or Stroyan and Luxemburg. ![]() ![]() In this note we give the elementary ingredients of a strong differentiation based on Abraham Robinson’s theory of infinitesimals. Primary 46G05 Secondary 02H25, 26A98, 58C20Ībstract: Differential calculus on nonnormed locally convex spaces suffers from technical difficulties (and the subsequent plethora of different definitions) partly because the families of multilinear maps over the spaces do not inherit a suitable topology. Infinitesimal calculus on locally convex spaces.
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