Showing papers on "Dual norm published in 1980"
TL;DR: In this paper, it was shown that even if the spectra of the operators A, B are contained in the segment [0, 1] the operator norm ∣A-B ∣ is arbitrarily small.
Abstract: One proves that
, where
is a segment containing the spectra of the self-adjoint operators A and B,
is the Lipschitz constant of the function
, and ∣·∣ is the operator norm. It is shown on examples that an estimate of the type
cannot be true for all
with
even if the spectra of the operators A, B are contained in the segment [0, 1] the norm ∣A–B∣ is arbitrarily small.
11 citations
TL;DR: In this paper, the interrelationships between norm convergence and two forms of convergence defined in terms of order, namely order and relative uniform convergence, are considered and the implications between conditions such as uniform convexity, uniform strictness, uniform monotonicity and others are proved.
Abstract: The interrelationships between norm convergence and two forms of convergence defined in terms of order, namely order and relative uniform convergence are considered. The implications between conditions such as uniform convexity, uniform strictness, uniform monotonicity and others are proved. In particular it is shown that a cr-order continuous, cr-order complete Banach lattice is order continuous. 1980 Mathematics subject classification (Amer. Math. Soc): primary 46 A 40.
8 citations
4 citations
4 citations
TL;DR: In this paper, the existence of extensions of continuous linear functionals from linear subspaces to the whole space, with arbitrarily prescribed larger norm, was proved in normed linear spaces, and under an additional boundedness assumption, in the known separation theorems for convex sets, there exist hyperplanes which separate and support both sets.
Abstract: We prove, in normed linear spaces, the existence of extensions of continuous linear functionals from linear subspaces to the whole space, with arbitrarily prescribed larger norm. Also, we prove that under an additional boundedness assumption, in the known separation theorems for convex sets, there exist hyperplanes which separate and support both sets.
4 citations
Journal Article•
TL;DR: In this article, it was shown that the p-local unconditional structure (p-l.u.st) is the largest possible theoretical value min{n 1/2, np}.
Abstract: There are Banach spaces which fail to have p-local unconditional structure (p-l.u.st.) for any p, ~ > p > 0. In particular, there exist n-dimensional Banach spaces En, n = 1, 2,..., whose p-l.u.st. constants are \"almost\" the largest possible theoretical value min{n1/2, np}. The p-l.u.st. constant is smaller and not equivalent to the usual l.u.st. constant.
4 citations
4 citations
TL;DR: In this article, a characterisation theory for best simultaneous approximation of a set of complex-valued bounded functions on a compact topological space B in a normed vector space, by elements of a non-linear subset of C(B) was presented.
2 citations
TL;DR: In this article, a generalization of the notion of the norm of a linear functional function is presented, and some applications of this generalization are discussed, including the case of linear functions.
1 citations
01 Jan 1980
TL;DR: In this article, the authors investigated Patil type approximations of functions f ∈ 1(D) with respect to the Hardy norm in the hardy space ℋ 1 (D).
Abstract: This chapter discusses approximation in the hardy space ℋ 1 (D) with respect to the norm topology. It focuses on investigating Patil type approximations of functions f ∈ 1(D) with respect to the Hardy norm. As there does not exist any continuous projector of L 1 (T) onto 1(T) by Newman's theorem, norm continuous Toeplitz operators on ℋ 1 (D) with symbols merely belonging to L 1 ∞ (T) are not available. The main tools are Toeplitz operators 1(T) → 1(T) with symbols that are smooth enough to multiply the strong dual BMO (T). The functions Re and Im are multipliers of the space BMO (T) for all t ∈ 0, 1. The Toeplitz operators are continuous endomorphisms of the complex Banach space ℋ 1 (T) with respect to the norm topology.