The hypertangent cone (original) (raw)
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On Polar Cones and Differentiability in Reflexive Banach Spaces
Let X be a Banach space, C ⊂ X be a closed convex set included in a well-based cone K, and also let σC be the support function which is defined on C. In this note, we first study the existence of a bounded base for the cone K, then using the obtained results, we find some geometric conditions for the set C, so that int(domσC) ̸ = ∅. The latter is a primary condition for subdifferen-tiability of the support function σC. Eventually, we study Gateaux differentiability of support function σC on two sets, the polar cone of K and int(domσC).
Methods of Functional Analysis and Topology, 2014
we can see: In a locally convex vector space E a barrel is defined to be an absolutely convex closed and absorbing subset A of E. The set U = {(a, b) ∈ E 2 , a − b ∈ A} then is seen to be a barrel in the sense of Roth's definition. With a counterexample, we show that it is not enough for U to be a barrel in the sense of Roth's definition. Then we correct this error with providing its converse and an application. 2000 Mathematics Subject Classification. 46A03.
A strong open mapping theorem for surjections from cones onto Banach spaces
Advances in Mathematics, 2014
We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael's Selection Theorem to yield the existence of a continuous bounded positively homogeneous right inverse of such a surjective map; a strong version of the usual Open Mapping Theorem is then a special case. As another consequence, an improved version of the analogue of Andô's Theorem for an ordered Banach space is obtained for a Banach space that is, more generally than in Andô's Theorem, a sum of possibly uncountably many closed not necessarily proper cones. Applications are given for a (pre)-ordered Banach space and for various spaces of continuous functions taking values in such a Banach space or, more generally, taking values in an arbitrary Banach space that is a finite sum of closed not necessarily proper cones.
On properties of subdifferential mappings in Frechet spaces
2005
UDC 517.9 We present conditions under which the subdifferential of a proper convex lower-semicontinuous functional in a Fréchet space is a bounded upper-semicontinuous mapping. The theorem on the boundedness of a subdifferential is also new for Banach spaces. We prove a generalized Weierstrass theorem in Fréchet spaces and study a variational inequality with a set-valued mapping. In recent years, operator and differential operator inclusions, multivariational inequalities, and systems containing both evolutionary and operator inclusions have been extensively studied. In reflexive Banach spaces, these objects were investigated by many authors (in particular, in [1-4]). Variational inequalities with a convex proper lower-semicontinuous functional ϕ are among the sources that generate operator inclusions [5]. In Banach spaces, the subdifferential ∂ϕ (⋅) of a proper lower-semicontinuous functional possesses several important properties [6-8], which play the key role in the investigation of variational inequalities. However, in separated locally convex spaces, analogous properties have not been investigated.