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CONTENTSChapter 0...............................................................................................................................................................................5   0.1. Introduction..................................................................................................................................................................5   0.2. Preliminary results.......................................................................................................................................................9Chapter...

In this note we present an affirmative answer to the problem posed by M. Baronti and C. Franchetti (oral communication) concerning a characterization of Lp-spaces among Orlicz sequence spaces. In fact, we show a more general characterization of Orlicz spaces isometric to Lp-spaces.

Let $X$ be a finite dimensional Banach space and let $Y\subset X$ be a hyperplane. Let $\text{L}{}_Y=\{L\in \text{L}(X,Y):L{\mid }_Y=0\}$. In this note, we present sufficient and necessary conditions on $L_0\in \text{L}{}_Y$ being a strongly unique best approximation for given $L\in \text{L}\left(X\right)$. Next we apply this characterization to the case of $X=l_{\infty }^n$ and to generalization of Theorem I.1.3 from [12] (see also [13]).

Let V be a two-dimensional real symmetric space with unit ball having 8n extreme points. Let λ(V) denote the absolute projection constant of V. We show that $\lambda \left(V\right)\le \lambda \left(V_n\right)$ where $V_n$ is the space whose ball is a regular 8n-polygon. Also we reprove a result of [1] and [5] which states that $4/\pi =\lambda \left(l{₂}^{\left(2\right)}\right)\ge \lambda \left(V\right)$ for any two-dimensional real symmetric space V.

We construct k-dimensional (k ≥ 3) subspaces $V^k$ of $l_1$, with a very simple structure and with projection constant satisfying $\lambda \left(V^k\right)\ge \lambda (V^k,l_1)>\lambda \left(l_2^{\left(k\right)}\right)$.

We say that a function from $X=C^L[0,1]$ is k-convex (for k ≤ L) if its kth derivative is nonnegative. Let P denote a projection from X onto V = Πₙ ⊂ X, where Πₙ denotes the space of algebraic polynomials of degree less than or equal to n. If we want P to leave invariant the cone of k-convex functions (k ≤ n), we find that such a demand is impossible to fulfill for nearly every k. Indeed, only for k = n-1 and k = n does such a projection exist. So let us consider instead a more general “shape” to preserve....

A theorem of Rudin permits us to determine minimal projections not only with respect to the operator norm but with respect to various norms on operator ideals and with respect to numerical radius. We prove a general result about N-minimal projections where N is a convex and lower semicontinuous (with respect to the strong operator topology) function and give specific examples for the cases of norms or seminorms of p-summing, p-integral and p-nuclear operator ideals.

Let V be an n-dimensional real Banach space and let λ(V) denote its absolute projection constant. For any N ∈ N with N ≥ n define $\lambda {ₙ}^N=sup\lambda \left(V\right):dim\left(V\right)=n,V\subset l_{\infty }^{\left(N\right)}$, λₙ = supλ(V): dim(V) = n. A well-known Grünbaum conjecture [Trans. Amer. Math. Soc. 95 (1960)] says that λ₂ = 4/3. König and Tomczak-Jaegermann [J. Funct. Anal. 119 (1994)] made an attempt to prove this conjecture. Unfortunately, their Proposition 3.1, used in the proof, is incorrect. In this paper a complete proof of the Grünbaum conjecture is presented

Let $Y\subset l_{\infty }^{\left(n\right)}$ be a hyperplane and let $A\in ℒ\left(Y\right)$ be given. Denote $$\begin{array}{ccc}\hfill 𝒜=& \{L\in ℒ(l_{\infty }^{\left(n\right)},Y):L\mid Y=A\}\text{and}\hfill & \hfill {\lambda }_A=inf\{\parallel L\parallel :L\in 𝒜\}.\end{array}$$ In this paper the problem of calculating of the constant ${\lambda }_A$ is studied. We present a complete characterization of those $A\in ℒ\left(Y\right)$ for which ${\lambda }_A=\parallel A\parallel$. Next we consider the case ${\lambda }_A>\parallel A\parallel$. Finally some computer examples will be presented.

In this paper we give a characterization of $\sigma$-order continuity of modular function spaces $L_{\varrho }$ in terms of the existence of best approximants by elements of order closed sublattices of $L_{\varrho }$. We consider separately the case of Musielak–Orlicz spaces generated by non-$\sigma$-finite measures. Such spaces are not modular function spaces and the proofs require somewhat different methods.