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Functionals of Bounded Frechet Variation

Published online by Cambridge University Press:  20 November 2018

Marston Morse  and
William Transue
Marston Morse
Affiliation:
The Institute for Advanced Study Kenyon College
William Transue
Affiliation:
The Institute for Advanced Study Kenyon College
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Extract

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In a series of papers which will follow this paper the authors will present a theory of functionals which are bilinear over a product A × B of two normed vector spaces A and B. This theory will include a representation theory, a variational theory, and a spectral theory. The associated characteristic equations will include as special cases the Jacobi equations of the classical variational theory when n = 1, and self-adjoint integrodifferential equations of very general type. The bilinear theory is oriented by the needs of non-linear and non-bilinear analysis in the large.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 1 , Issue 2 , 01 April 1949 , pp. 153 - 165
Copyright
Copyright © Canadian Mathematical Society 1949

References

1 See Banach, Théorie des opérations linéaires (Warsaw, 1932), chap. IV.

2 Morse and Transue, “Functionals F Bilinear over the Produce A × B of Two Pseudonormed Vector Spaces. I. The Representation of F,” Ann, of Math, (To be published.)

3 Fréchet, , “Sur les fonctionnelles bilinéaires,” Trans. Amer. Math. Soc, vol. 16 (1915), 215-234.Google Scholar In this basic memoir Fréchet obtains a representation of any functional K which is bilinear on C × C, in the form of a repeated Stieltjeas integral with a distribution function k of the above type.

4 Clarkson and Adams, “On Definitions of Bounded Variation for Functions of Two Variables,” Trans. Amer. Math. Soc., vol. 35 (1933), 824-854.

5 The equality holds in (3.5).

6 McShane, , Integration (Princeton, 1944), 41.Google Scholar