Abstract
This is a service chapter. Here we shall build a new technical machinery to work with regularity problems for mappings between Banach spaces.
This is a preview of subscription content, log in via an institution to check access.
Notes
- 1.
In the literature the term “Gâteaux smooth space” is sometimes used for a broader class of spaces in which every convex function is Gâteaux differentiable on a dense set [117].
- 2.
Recall that, given a set T, the space \(c_0(T)\) is defined as the subspace of \(\ell _{\infty }(T)\) consisting of all functions f such that for any \(\varepsilon >0\) the set \([|f|>\varepsilon ]\) is at most finite.
- 3.
Note that, with few exceptions (e.g. generalized gradient of non-Lipschitz functions), the equality holds for known subdifferentials.
- 4.
“Subderivative" may be a more correct term as we usually do not work in Hilbert spaces. But, due to a tradition going back to convex analysis, the term “subdifferential" is typically used in the literature.
- 5.
For this reason the subdifferentials defined in this section are sometimes (and naturally) called canonical. There is another approach to subdifferentiation which is widely used in the theory of viscosity solutions to differential equations. For Fréchet subdifferentials both lead to equivalent definitions (see e.g. [79]).
- 6.
This collection may differ from \({\mathcal {S}}(X)\) because the subspace A(E) may not be closed in Y for some \(E\in {\mathcal {S}}(X)\).
- 7.
As a matter of fact, we shall see in Chap. 7 that the most general versions of all basic rules of the subdifferential calculus are simple consequences of some elementary principles of regularity theory.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Ioffe, A.D. (2017). Subdifferentials: A Short Introduction. In: Variational Analysis of Regular Mappings. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-64277-2_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-64277-2_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-64276-5
Online ISBN: 978-3-319-64277-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)