This paper is about certain linear subspaces of Banach SN spaces (that is to say Banach spaces which have a symmetric nonexpansive linear map into their dual spaces). We apply our results to monotone linear subspaces of the product of a Banach space and its dual. In this paper, we establish several new results and also give improved proofs of some known ones in both the general and the special contexts.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Arens, R.: Operational calculus of linear relations. Pac. J. Math. 11, 9–23 (1961)
Brezis, H., Browder, F.E.: Linear maximal monotone operators and singular nonlinear integral equations of Hammerstein, type. In: Nonlinear Analysis (Collection of Papers in Honor of Erich H. Rothe), pp. 31–42. Academic Press, New York (1978)
Bauschke, H., Borwein, J.M., Wang, X., Yao, L.: For maximally monotone linear relations, dense type, negative-infimum type, and Fitzpatrick–Phelps type all coincide with monotonicity of the adjoint. http://arxiv.org/abs/1103.6239v1. Posted 31 Mar 2011
Bauschke, H., Borwein, J.M., Wang, X., Yao, L.: Every maximally monotone operator of Fitzpatrick–Phelps type is actually of dense type. Optim. Lett. 6, 1875–1881 (2012). doi:10.1007/s11590-011-0383-2
Bauschke, H., Borwein, J.M., Wang, X., Yao, L.: The Brezis–Browder Theorem in a general Banach space. J. Funct. Anal. 262, 4948–4971 (2012). doi:10.1016/j.jfa.2012.03.023
Brøndsted, A., Rockafellar, R.T.: On the subdifferentiability of convex functions. Proc. Am. Math. Soc. 16, 605–611 (1965)
García Ramos, Y., Martínez-Legaz, J.E., Simons, S.: New results on q–positivity. Positivity 16, 543–563 (2012). doi:10.1007/s11117-012-0191-7
Kelley, J.L., Namioka, I., co-authors: Linear Topological Spaces. D. Van Nostrand Co., Inc., Princeton/Toronto/London/Melbourne (1963)
Marques Alves, M., Svaiter, B.F.: On Gossez type (D) maximal monotone operators. J. Convex. Anal. 17, 1077–1088 (2010)
Rockafellar, R.T.: Extension of Fenchel’s duality theorem for convex functions. Duke Math. J. 33, 81–89 (1966)
Simons, S.: The range of a monotone operator. J. Math. Anal. Appl. 199, 176–201 (1996)
Simons, S.: From Hahn–Banach to Monotonicity, 2nd edn. Lecture Notes in Mathematics, vol. 1693. Springer–Verlag (2008)
Simons, S.: Banach SSD spaces and classes of monotone sets. J. Convex. Anal. 18, 227–258 (2011)
Simons, S.: Linear L–positive sets and their polar subspaces. Set-Valued Var. Anal. 20, 603–615 (2012). doi:10.1007/s11228-012-0206-3
Yao, L.: The Brezis–Browder theorem revisited and properties of Fitzpatrick functions of order n. Fixed-point algorithms for inverse problems in science and engineering. Springer Optim. Appl. 49, 391–402 (2011)
Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific (2002)
About this article
Cite this article
Simons, S. Polar Subspaces and Automatic Maximality. Set-Valued Var. Anal 22, 259–270 (2014). https://doi.org/10.1007/s11228-013-0244-5
- Banach space
- Dual and bidual
- Maximally monotone set
- Type (NI)
- Subdifferential of a convex function
- Brezis–Browder theorem
- Rockafellar’s formula for the subdifferential of a sum
- Brondsted–Rockafellar theorem
- Polar subspace
Mathematics Subject Classifications (2010)