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On multilinear mappings attaining their norms.

Maria Acosta — 1998

Studia Mathematica

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.

Denseness of norm attaining mappings.

María D. Acosta — 2006

RACSAM

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...

Shilov boundary for holomorphic functions on some classical Banach spaces

María D. AcostaMary Lilian Lourenço — 2007

Studia Mathematica

Let ( B X ) be the Banach space of all bounded and continuous functions on the closed unit ball B X of a complex Banach space X and holomorphic on the open unit ball, with sup norm, and let u ( B X ) be the subspace of ( B X ) of those functions which are uniformly continuous on B X . A subset B B X is a boundary for ( B X ) if f = s u p x B | f ( x ) | for every f ( B X ) . 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 ( B X ) . On the other hand, for X = , the Schreier space,...

Norm attaining and numerical radius attaining operators.

María D. AcostaRafael Payá — 1989

Revista Matemática de la Universidad Complutense de Madrid

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...

Dual spaces generated by the interior of the set of norm attaining functionals

Maria D. AcostaJulio Becerra GuerreroManuel Ruiz Galán — 2002

Studia Mathematica

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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