We show, for any Banach spaces X and Y, the denseness of the set of bilinear forms on X × Y whose third Arens transpose attains its norm. We also prove the denseness of the set of norm attaining multilinear mappings in the class of multilinear mappings which are weakly continuous on bounded sets, under some additional assumptions on the Banach spaces, and give several examples of classical spaces satisfying these hypotheses.
The Bishop-Phelps Theorem states that the set of (bounded and linear) functionals on a Banach space that attain their norms is dense in the dual. In the complex case, Lomonosov proved that there may be a closed, convex and bounded subset C of a Banach space such that the set of functionals whose maximum modulus is attained on C is not dense in the dual. This paper contains a survey of versions for operators, multilinear forms and polynomials of the Bishop-Phelps Theorem. Lindenstrauss provided examples...
Let be the Banach space of all bounded and continuous functions on the closed unit ball of a complex Banach space X and holomorphic on the open unit ball, with sup norm, and let be the subspace of of those functions which are uniformly continuous on . A subset is a boundary for if for every . We prove that for X = d(w,1) (the Lorentz sequence space) and X = C₁(H), the trace class operators, there is a minimal closed boundary for . On the other hand, for X = , the Schreier space,...
In this note we deal with a version of James' Theorem for numerical radius, which was already considered in [4]. First of all, let us recall that this well known classical result states that a Banach space satisfying that all the (bounded and linear) functionals attain the norm, has to be reflexive [16].
In this note we discuss some results on numerical radius attaining operators paralleling earlier results on norm attaining operators. For arbitrary Banach spaces X and Y, the set of (bounded, linear) operators from X to Y whose adjoints attain their norms is norm-dense in the space of all operators. This theorem, due to W. Zizler, improves an earlier result by J. Lindenstrauss on the denseness of operators whose second adjoints attain their norms, and is also related to a recent result by C. Stegall...
We characterize some isomorphic properties of Banach spaces in terms of the set of norm attaining functionals. The main result states that a Banach space is reflexive as soon as it does not contain ℓ₁ and the dual unit ball is the w*-closure of the convex hull of elements contained in the "uniform" interior of the set of norm attaining functionals. By assuming a very weak isometric condition (lack of roughness) instead of not containing ℓ₁, we also obtain a similar result. As a consequence of the...
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