arXiv subjects

Nikica Uglesic

Publications and source records attributed to Nikica Uglesic.

3 recordsLinked to original sources

arXiv2019

An extension of the normed dual functors

By means of the direct limit technique, with every normed space X it is associated a bidualic (Banach) space $\tilde{X} (D^2( \tilde{X}) \cong \tilde{X} $ - called the hyperdual of $X$) that contains (isometrically embedded) $X$ as well as all the even (normed) duals $D^{2n}(X)$, which make an increasing sequence of the category retracts. The algebraic dimension dim $\tilde{X}$ = dim $X$ (dim $\tilde{X}$ = $2^{\aleph_0}$ ), whenever dim $X \neq \aleph_0$, (dim $X = \aleph_0$). Furthermore, the correspondence $X \mapsto \tilde{X}$ extends to a faithful covariant functor (called the hyperdual functor) on the category of normed spaces.

math.FA↗

arXiv2019

The quotient shapes of normed spaces and application

The quotient shape types of normed vectorial spaces(over the same field) with respect to Banach spaces reduce to those of Banach spaces. The finite quotient shape type of normed spaces is an invariant of the (algebraic) dimension, but not conversely. The converse holds for separable normed spaces as well as for the bidual-like spaces (isomorphic to their second dual spaces). As a consequence, the Hilbert space $l_2$, or even its (countably dimensional, unitary) direct sum subspace may represent the unique quotient shape type of all $2^\aleph_0$-dimensional normed spaces. An application yields two extension type theorems.

math.FA↗

arXiv2019

On the duals of normed spaces and quotient shapes

Some properties of the (normed) dual Hom-functor $D$ and its iterations $D^n$ are exhibited. For instance: $D$ turns every canonical embedding (in the second dual space) into a retraction (of the third dual onto the first one); $D$ rises the countably infinite (algebraic) dimension only; $D$ does not change the finite quotient shape type. By means of that, the finite quotient shape classification of normed vectorial spaces is completely solved. As a consequence, two extension type theorems are derived.

math.FA↗