Uniqueness of minimal projections in smooth expanded matrix spaces

Let us consider the space M(n, m) of all real or complex matrices on n rows and m columns. In 2000 Lesław Skrzypek proved the uniqueness of minimal projection of this space onto its subspace $$M(n,1)+M(1,m)$$ M ( n , 1 ) + M ( 1 , m ) which consists of all sums of matrices with constant rows and matrices with constant columns. We generalize this result using some new methods proved by Lewicki and Skrzypek (J Approx Theory 148:71–91, 2007). Let S be a space of all functions from $$X\times Y \times Z$$ X × Y × Z into $${\mathbb {R}}$$ R or $${\mathbb {C}}$$ C , where X, Y, Z are finite sets. It could be interpreted as a space of three-dimensional matrices M(n, m, r). Let T be a subspace of S consisting of all sums of functions which depend on one variable. Let S be equipped with a smooth norm $$\Vert .\Vert $$ ‖ . ‖ . We show that there exists the unique minimal projection of S onto its subspace T.

Some topics in differential geometry of normed spaces

For a surface immersed in a three-dimensional space endowed with a smooth norm instead of an inner product, one can define analogous concepts of curvature and metric. With such concepts in mind, various questions immediately appear. The aim of this paper is to propose and answer some of those questions. In this framework we prove several characterizations of minimal surfaces in normed spaces, and respective analogues of some global theorems (e.g., Hadamard-type theorems) are also derived. A result on the curvature of surfaces having constant Minkowskian width is given, and finally we study the ambient metric induced on the surface, proving an extension of the classical Bonnet theorem.

Smoothness, Convexity, Porosity, and Separable Determination

This chapter shows how spaces with separable dual admit a Fréchet smooth norm. It first considers a criterion of the differentiability of continuous convex functions on Banach spaces before discussing Fréchet smooth and nonsmooth renormings and Fréchet differentiability of convex functions. It then describes the connection between porous sets and Fréchet differentiability, along with the set of points of Fréchet differentiability of maps between Banach spaces. It also examines the concept of separable determination, the relevance of the σ‎-porous sets for differentiability and proves the existence of a Fréchet smooth equivalent norm on a Banach space with separable dual. The chapter concludes by explaining how one can show that many differentiability type results hold in nonseparable spaces provided they hold in separable ones.

Fr ´Echet Differentiability of Vector-Valued Functions

This chapter shows that if a Banach space with a Fréchet smooth norm is asymptotically smooth with modulus o(tⁿ logⁿ⁻¹(1/t)) then every Lipschitz map of X to a space of dimension not exceeding n has many points of Fréchet differentiability. In particular, it proves that two real-valued Lipschitz functions on a Hilbert space have a common point of Fréchet differentiability. The chapter first presents the theorem whose assumptions hold for any space X with separable dual, includes the result that real-valued Lipschitz functions on such spaces have points of Fréchet differentiability, and takes into account the corresponding mean value estimate. The chapter then gives the estimate for a “regularity parameter” and reduces the theorem to a special case. Finally, it discusses simplifications of the arguments of the proof of the main result in some special situations.

BMO Functions and Carleson Measures with Values in Uniformly Convex Spaces

This paper studies the relationship between vector-valued BMO functions and the Carleson measures defined by their gradients. Let dA and dm denote Lebesgue measures on the unit disc D and the unit circle 𝕋, respectively. For 1 < q < ∞ and a Banach space B, we prove that there exists a positive constant c such thatholds for all trigonometric polynomials f with coefficients in B if and only if B admits an equivalent norm which is q-uniformly convex, whereThe validity of the converse inequality is equivalent to the existence of an equivalent q-uniformly smooth norm.

Approximation by functions with bounded derivative on Banach spaces

Let X be a separable Banach space which admits a C1-smooth norm, and let G ⊂ X be an open subset. Then any real-valued, bounded and uniformly continuous map on G can be uniformly approximated on G by C1-smooth functions with bounded derivative.

SMOOTH LIPSCHITZ RETRACTIONS OF STARLIKE BODIES ONTO THEIR BOUNDARIES IN INFINITE-DIMENSIONAL BANACH SPACES

Let X be an infinite-dimensional Banach space, and let A be a Cp Lipschitz bounded starlike body (for instance the unit ball of a smooth norm). We prove that:(1) the boundary ∂A is Cp Lipschitz contractible;(2) there is a Cp Lipschitz retraction from A onto ∂A;(3) there is a Cp Lipschitz map T : A → A with no approximate fixed points.

Structure of 6H–SiC(0001) Surfaces from AB-Initio Calculations

We report the results of ab-initio calculations of structural properties of hexagonal 6H–SiC(0001) surfaces. The calculations have been carried out self-consistently within local density approximation employing supercell geometries, smooth norm-conserving pseudopotentials in separable form and Gaussian orbital basis sets. We have investigated several structural models for adatom-induced [Formula: see text] reconstructions with adsorbed Si or C adatoms or trimers residing in threefold-symmetric T 4 or H 3 positions above Si- or C-terminated substrate surfaces, respectively. In the case of the Si-terminated substrate surface our results favor Si adatoms in T 4 sites as optimal configuration in very good agreement with experimental data. For the C-terminated substrate surface our results indicate that none of the investigated [Formula: see text] adatom or trimer configurations is the optimal surface structure.