Related papers: An extension of the normed dual functors

Sequences of $n$-th order bidual Banach spaces, called tower systems and their direct and inverse limits are considered. Motivated by recent applications in uniform algebras, we introduce two functors: $\textrm{Dir}$ and $\textrm{Inv}$…

For a real normed space $X$, we study the $n$-dual space of $\left(X,\left\Vert \cdot \right\Vert \right) $ and show that the space is a Banach space. Meanwhile, for a real normed space $X$ of dimension $d\geq n$ which satisfies property…

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…

Under certain hypotheses on the Banach space $X$, we prove that the set of analytic functions in $\mathcal{A}_u(X)$ (the algebra of all holomorphic and uniformly continuous functions in the ball of $X$) whose Aron-Berner extensions attain…

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…

We present a construction that enables one to find Banach spaces $X$ whose sets $NA(X)$ of norm attaining functionals do not contain two-dimensional subspaces and such that, consequently, $X$ does not contain proximinal subspaces of finite…

The classical theorem of Bishop-Phelps asserts that, for a Banach space X, the norm-achieving functionals in X* are dense in X*. Bela Bollobas's extension of the theorem gives a quantitative description of just how dense the norm-achieving…

Let $X$ be a Banach space with the unit ball $B(X)$ and $A\subset X$ be a convex origin-symmetric compact in $X$. Let $\mathrm{j}:X\rightarrow \widetilde{X}$ be an isometric extension of $X$. It is well-known that linear widths $\lambda…

A Fenchel-Moreau type duality for proper convex and lower semi-continuous functions $f\colon X\to \overline{L^0}$ is established where $(X,Y,\langle \cdot,\cdot \rangle)$ is a dual pair of Banach spaces and $\overline{L^0}$ is the set of…

We propose a systematic construction of native Banach spaces for general spline-admissible operators ${\rm L}$. In short, the native space for ${\rm L}$ and the (dual) norm $\|\cdot\|_{\mathcal{X}'}$ is the largest space of functions $f:…

This paper deals with some basic constructions of linear and multilinear algebra on finite-dimensional diffeological vector spaces. We consider the diffeological dual formally checking that the assignment to each space of its dual defines a…

Let $X=\bigcup\varphi_{i}X$ be a strongly separated self-affine set in $\mathbb{R}^2$ (or one satisfying the strong open set condition). Under mild non-compactness and irreducibility assumptions on the matrix parts of the $\varphi_{i}$, we…

We prove that, any problem of minimization of proper lower semicontinuous function defined on a normal Hausdorff space, is canonically equivalent to a problem of minimization of a proper weak * lower semicontinuous convex function defined…

A Banach space $X$ has the $2$-summing property if the norm of every linear operator from $X$ to a Hilbert space is equal to the $2$-summing norm of the operator. Up to a point, the theory of spaces which have this property is independent…

The dimension algebra of graded groups is introduced. With the help of known geometric results of extension theory that algebra induces all known results of the cohomological dimension theory. Elements of the algebra are equivalence classes…

We show that if the Szlenk index of a Banach space $X$ is larger than the first infinite ordinal $\omega$ or if the Szlenk index of its dual is larger than $\omega$, then the tree of all finite sequences of integers equipped with the…

Let X be a separable metric space and let \beta be the strict topology on the space of bounded continuous functions on X, which has the space of \tau-additive Borel measures as a continuous dual space. We prove a Banach-Dieudonne\'{e} type…

In this note, we show that if a Banach space X has a predual, then every bounded linear operator on X with a continuous functional calculus admits a bounded Borel functional calculus. A consequence of this is that on such a Banach space,…

For a complex Banach space $\mathbb X$, we prove that $\mathbb X$ is a Hilbert space if and only if every strict contraction $T$ on $\mathbb X$ dilates to an isometry if and only if for every strict contraction $T$ on $\mathbb X$ the…

We prove that every separable Banach space containing $\ell_1$ can be equivalently renormed so that its bidual space is octahedral, which answers, in the separable case, a question by Godefroy in 1989. As a direct consequence, we obtain…