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A new type of orthogonality for normed planes

Horst Martini, Margarita Spirova (2010)

Czechoslovak Mathematical Journal

In this paper we introduce a new type of orthogonality for real normed planes which coincides with usual orthogonality in the Euclidean situation. With the help of this type of orthogonality we derive several characterizations of the Euclidean plane among all normed planes, all of them yielding also characteristic properties of inner product spaces among real normed linear spaces of dimensions d 3 .

A theorem on isotropic spaces

Félix Cabello Sánchez (1999)

Studia Mathematica

Let X be a normed space and G F ( X ) the group of all linear surjective isometries of X that are finite-dimensional perturbations of the identity. We prove that if G F ( X ) acts transitively on the unit sphere then X must be an inner product space.

Banach spaces which embed into their dual

Valerio Capraro, Stefano Rossi (2011)

Colloquium Mathematicae

We use Birkhoff-James' orthogonality in Banach spaces to provide new conditions for the converse of the classical Riesz representation theorem.

Directionally Euclidean structures of Banach spaces

Jarno Talponen (2011)

Studia Mathematica

We study Banach spaces with directionally asymptotically controlled ellipsoid-approximations of the unit ball in finite-dimensional sections. Here these ellipsoids are the unique minimum volume ellipsoids, which contain the unit ball of the corresponding finite-dimensional subspace. The directional control here means that we evaluate the ellipsoids by means of a given functional of the dual space. The term 'asymptotical' refers to the fact that we take 'lim sup' over finite-dimensional subspaces. ...

Espacios de producto interno (II).

Palaniappan Kannappan (1995)

Mathware and Soft Computing

Among normal linear spaces, the inner product spaces (i.p.s.) are particularly interesting. Many characterizations of i.p.s. among linear spaces are known using various functional equations. Three functional equations characterizations of i.p.s. are based on the Frchet condition, the Jordan and von Neumann identity and the Ptolemaic inequality respectively. The object of this paper is to solve generalizations of these functional equations.

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