Abstract

Let YY be a subspace of a real normed space XX. We say that the couple (X,Y)(X,Y) has the CE\mathrm{CE}-property (``convex extension property'') if each continuous convex function on YY admits a continuous convex extension defined on XX. By using techniques of Johnson and Zippin, we prove the following results about the CE\mathrm{CE}-property: if XX is the c0(Γ)c_0(\Gamma)-sum or the ℓp(Γ)\ell_p(\Gamma)-sum (1<p<∞1<p<\infty) of separable normed spaces, then the couple (X,Y)(X,Y) has the CE\mathrm{CE}-property, for each subspace YY of XX. Another similar result concerns weak∗^*-closed subspaces YY of X=ℓ1(Γ)=c0(Γ)∗X=\ell_1(\Gamma)=c_0(\Gamma)^*.

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Keywords

  • Convex function
  • extension
  • normed linear space

Mathematics Subject Classification

52A41; 26B25, 46A99

References

  1. A. Avilés, F. Cábello Sanchez, J. M. F. Castillo, M. Gonzáles, Y. Moreno: On separably injective Banach spaces, Adv. Math. 234 (2013) 192–216.https://doi.org/10.1016/j.aim.2012.10.013Google Scholar
  2. P. Bandyopadhyay, G. Godefroy: Linear structures in the set of norm-attaining functionals on a Banach space, J. Convex Analysis 13 (2006) 489–497.https://doi.org/10.68381/jca13042Google Scholar
  3. J. Borwein, V. Montesinos, J. Vanderwerff: Boundedness, differentiability and extensions of convex functions, J. Convex Analysis 13 (2006) 587–602.https://doi.org/10.68381/jca13046Google Scholar
  4. C. A. De Bernardi, L. Veselý: Extension of continuous convex functions from subspaces I, J. Convex Analysis 21 (2014) 1065–1084.https://doi.org/10.68381/jca21057Google Scholar
  5. C. A. De Bernardi, L. Veselý: Extension of continuous convex functions from subspaces II, J. Convex Analysis 22 (2015) 101–116.https://doi.org/10.68381/jca22006Google Scholar
  6. M. Fabian, P. Habala, P. Hájek, V. Montesinos, V. Zizler: Banach Space Theory. The Basis for Linear and Nonlinear Analysis, CMS Books in Mathematics, Springer, New York (2011).https://doi.org/10.1007/978-1-4419-7515-7Google Scholar
  7. K. Floret: Weakly Compact Sets, Lecture Notes in Mathematics 801, Springer, Berlin (1980).https://doi.org/10.1007/BFb0091483Google Scholar
  8. W. B. Johnson, M. Zippin: Extension of operators from subspaces of c0 (Γ) into C(K) spaces, Proc. Amer. Math. Soc. 107 (1989) 751–754.https://doi.org/10.1090/S0002-9939-1989-0984799-7Google Scholar
  9. H. P. Rosenthal: The complete separable extension property, J. Oper. Theory 43 (2000) 329–374.Google Scholar
  10. L. Veselý, L. Zajíček: On extensions of d.c. functions and convex functions, J. Convex Analysis 17 (2010) 427–440.https://doi.org/10.68381/jca17030Google Scholar
  11. M. Zippin: Extension of bounded linear operators, in: Handbook of the Geometry of Banach Spaces, Vol. II, W. B. Johnson et al. (ed.), North-Holland, Amsterdam (2003) 1703–1741.https://doi.org/10.1016/S1874-5849(03)80047-5Google Scholar

Suggested citation

C. A. De Bernardi. “A Note on the Extension of Continuous Convex Functions from Subspaces.” Journal of Convex Analysis 24 (2017), No. 1, 333–347. https://doi.org/10.68381/jca24023

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Published by Heldermann Verlag, 2017. Rights now held by Banach Press LLC.