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Two Unexpected Examples Concerning Differentiability of Lipschitz Functions on Banach Spaces

  • David Preiss
  • Jaroslav Tišer
Conference paper
Part of the Operator Theory Advances and Applications book series (OT, volume 77)

Abstract

In this note we present two examples illustrating some surprising relations between the known existence theorems concerning two main concepts of differentiability of Lipschitz functions between Banach spaces. Recall that these concepts are: Gâteaux derivative of a mapping φ: XY at xX, which is defined as a continuous linear map φ′ (x): XY verifying
$$ \left\langle {\varphi '(x),u} \right\rangle = \mathop {\lim }\limits_{t \to 0} \frac{{\phi \left( {x + tu} \right) - \phi (x)}}{t} $$
for every uX, and Fréchet derivative which, in addition, requests that the above limit be uniform for ∥u∥ ≤ 1.

Keywords

Lipschitz Function Gaussian Measure Countable Union Continuous Linear Mapping Closed Convex Hull 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 1995

Authors and Affiliations

  • David Preiss
    • 1
  • Jaroslav Tišer
    • 2
  1. 1.Department of MathematicsUniversity College LondonLondonUK
  2. 2.Department of MathematicsCzech Technical University Faculty of Electrical EngineeringPrahaCzech Republic

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