Calculus in Banach Spaces

  • Ward Cheney
Part of the Graduate Texts in Mathematics book series (GTM, volume 208)


In this chapter we develop the theory of the derivative for mappings between Banach spaces. Partial derivatives, Jacobians, and gradients are all examples of the general theory, as are the Gâteaux and Fréchet differentials. Kantorovich’s theorem on Newton’s method is proved. Following that there is a section on implicit function theorems in a general setting. Such theorems can often be used to prove the existence of solutions to integral equations and other similar problems. Another section, devoted to extremum problems, illustrates how the methods of calculus (in Banach spaces) can lead to solutions. A section on the “calculus of variations” closes the chapter.


Banach Space Lagrange Multiplier Euler Equation Extremum Problem Implicit Function Theorem 
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Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Ward Cheney
    • 1
  1. 1.Department of MathematicsUniversity of Texas at AustinAustinUSA

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