We prove a general result on complemented unconditional basic sequences in Banach lattices and apply it to give some new examples of spaces with unique unconditional basis. We show that Tsirelson space and certain Nakano spaces have unique unconditional bases. We also construct an example of a space with a unique unconditional basis with a complemented subspace failing to have a unique uncondit...

If X is a Banach space such that the isomorphism constant to `2 from n dimensional subspaces grows sufficiently slowly as n → ∞, then X has the approximation property. A consequence of this is that there is a Banach space X with a symmetric basis but not isomorphic to `2 so that all subspaces of X have the approximation property. This answers a problem raised in 1980 [8]. An application of the ...

The aim of this paper is to characterize one-complemented subspaces of finite codimension in the Musielak–Orlicz sequence space l . We generalize the well-known fact (Ann. Mat. Pura Appl. 152 (1988) 53; Period. Math. Hungar. 22 (1991) 161; Classical Banach Spaces I, Springer, Berlin, 1977) that a subspace of finite codimension in lp, 1 p<∞, is one-complemented if and only if it can be expressed...

It is proved that there exist complemented subspaces of countable topo-logical products (locally convex direct sums) of Banach spaces which cannot be represented as topological products (locally convex direct sums) of Banach spaces The problem of description of complemented subspaces of a given locally convex space is one of the general problems of structure theory of locally convex spaces. In ...

In this note we include two remarks about bounded (not necessarily contractive) linear projections on a von Neumann-algebra. We show that if M is a von Neumann-subalgebra of B(H) which is complemented in B(H) and isomorphic to M ⊗ M then M is injective (or equivalently M is contractively complemented). We do not know how to get rid of the second assumption on M. In the second part,we show that ...

Let n ≥ 3. From the description of subdirectly irreducible complemented Arguesian lattices with four generators given by Herrmann, Ringel and Wille it follows that the subspace lattice of an n-dimensional vector space over a finite field is generated by four elements if and only if the field is a prime field. By exhibiting a 5-element generating set we prove that the subspace lattice of an n-di...

A subspace V of a Banach space X is said to be complemented if there exists a (bounded) projection mapping X onto V . Obviously all subspaces of finite-dimension are complemented. The goal of this note is to show that there are (relatively) few monotonically complemented subspaces of finite-dimension in X = (C[a, b], ‖ · ‖∞); that is, finitedimensional subspaces V ⊂ X for which there exists a p...

In this paper we continue the study of central subspaces initiated in [2] and its infinite version called almost constrained subspaces. We are interested in studying situations where these intersection properties of balls lead to the existence of a linear projection of norm one. We show that every finite dimensional subspace is a central subspace only in Hilbert spaces. By considering direct su...

In this article, we consider pair frames in Banach spaces and   introduce Banach pair frames. Some various concepts in the frame theory such as frames, Schauder frames, Banach frames and atomic decompositions are considered as   special kinds of (Banach) pair frames.  Some frame-like inequalities  for (Banach)  pair frames are presented. The elements that participant  in the construction of (Ba...