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Given two Banach spaces X and Y , we introduce and study a concept of norm-attainment in the space of nuclear operators N (X, Y ) and in the projective tensor product space X ⊗ π Y . We exhibit positive and negative examples where both previous norm-attainment hold. We also study the problem of whether the class of elements which attain their norms in N (X, Y ) and in X ⊗ π Y is dense or not. We prove that, for both concepts, the density of norm-attaining elements holds for a large class of Banach spaces X and Y which, in particular, covers all classical Banach spaces. Nevertheless, we present Banach spaces X and Y failing the approximation property in such a way that the class of elements in X ⊗ π Y which attain their projective norms is not dense. We also discuss some relations and applications of our work to the classical theory of

Given a pointed metric space M, we study when there exist n-dimensional linear subspaces of $$\mathrm {Lip}_0(M)$$ Lip 0 ( M ) consisting of strongly norm-attaining Lipschitz functionals, for $$n\in {\mathbb {N}}$$ n ∈ N . We show that this is always the case for infinite metric spaces, providing a definitive answer to the question. We also study the possible sizes of such infinite-dimensional closed linear subspaces Y, as well as the inverse question, that is, the possible sizes for a metric space M in order to such a subspace Y exist. We also show that if the metric space M is $$\sigma $$ σ -precompact, then the aforementioned subspaces Y need to be always separable and isomorphically polyhedral, and we show that for spaces containing [0, 1] isometrically, they can be infinite-dimensional.

Resumen:Introducción: La incorporación de la Clínica y la Imagenología permiten una mejor comprensión de la Anatomía. El objetivo de este trabajo es desarrollar un prototipo rápido en material sintético que replique detalles anatómicos para ser utilizado en la docencia y el entrenamiento quirúrgico en Pediatría. Material y Método: Presentación de caso: Paciente de un año de edad con síndrome de dificultad respiratoria. En el examen endoscópico se halló una compresión traqueal distal. La angiotomografía confirmó la presencia de una malformación vascular.Con la finalidad de analizar una conducta adecuada, se solicitó la confección de un prototipo rápido a escala 1:1 que simulara una condición idéntica a la topografía torácica del paciente, utilizando imágenes virtuales 3D almacenadas en formato DICOM. Técnica de generación de prototipo rápido: Se obtuvo una malla digital tridimensional y se generó el código "g" que se utilizó para controlar el hardware de producción. Se efectuó simulación digital y producción en material plástico (ABS) con técnica de deposición y fusión (MDF). Se validó el prototipo comparándolo con las mediciones testigos del modelo virtual en 3 D. Resultados y Discusión: El modelo replicó exactamente los defectos hallados en la tomografía y endoscopía, confirmando la presencia de la malformación vascular y su repercusión sobre el aparato respiratorio. El prototipo rápido muestra las estructuras internas y externas del cuerpo humano con máxima precisión permitiendo una visión topográfica de situaciones "normales o patológicas" que facilitaría la docencia y el entrenamiento del equipo quirúrgico para proponer un plan de tratamiento adecuado. Hay numerosas áreas de la medicina que se beneficiarían con este modelo que podría ser construído con diversos tipos de materiales de diferente flexibilidad y consistencia.Conclusiones: El prototipo rápido le da estado físico a las imágenes virtuales 3D, permitiendo la docencia y entrenamiento del equipo quirúrgico. Palabras clave: prototipo rápido, enseñanza de pediatría y anatomía Abstract:Introduction: The incorporation of the clinic and the imaging allow a better understanding of anatomy. The aim of this work is to develop a rapid prototype in synthetic material that replicates anatomical details to be used in teaching and surgical training in Pediatrics. Material and method: presentation of case: one year old female with respiratory distress syndrome. In the endoscopic examination was found a distal tracheal compression. The angiotomography confirmed the presence of a vascular malformation. In order to discuss appropriate conduct, the making of rapid prototyping in scale 1:1 was requested to simulate an identical condition of the thoracic topography of the patient, using virtual 3D images stored in the DICOM format. Rapid prototype technique: code "g" was generated, which was used to control the hardware of production and a three-dimensional digital grid was obtained. Digital simulation and production in plastic (ABS) with deposition and fusion technique ...

In this paper, we are interested in studying the set A }¨} pX, Y q of all normattaining operators T from X into Y satisfying the following: given ε ą 0, there exists η such that if }T x} ą 1´η, then there is x 0 such that }x 0´x } ă ε and T itself attains its norm at x 0 . We show that every norm one functional on c 0 which attains its norm belongs to A }¨} pc 0 , Kq. Also, we prove that the analogous result holds neither for A }¨} p 1 , Kq nor A }¨} p 8 , Kq. Under some assumptions, we show that the sphere of the compact operators belongs to A }¨} pX, Y q and that this is no longer true when some of these hypotheses are dropped. The analogous set A nu pXq for numerical radius of an operator instead of its norm is also defined and studied. We present a complete characterization for the diagonal operators which belong to the sets A }¨} pX, Xq and A nu pXq when X " c 0 or p . As a consequence, we get that the canonical projections P N on these spaces belong to our sets. We give examples of operators on infinite dimensional Banach spaces which belong to A }¨} pX, Xq but not to A nu pXq and vice-versa. Finally, we establish some techniques which allow us to connect both sets by using direct sums.