We prove that the dual or any quotient of a separable reflexive Banach space with the unconditional tree property has the unconditional tree property. This is used to prove that a separable reflexive Banach space with the unconditional tree property embeds into a reflexive Banach space with an unconditional basis. This solves several long standing open problems. In particular, it yields that a ...

We show that when a linear quotient map to a separable Banach space X has a Lipschitz right inverse, then it has a linear right inverse. If a separable space X embeds isometrically into a Banach space Y , then Y contains an isometric linear copy of X. This is false for every nonseparable weakly compactly generated Banach space X. Canonical examples of nonseparable Banach spaces which are Lipsch...

we study the theory of best approximation in tensor product and the direct sum of some lattice normed spacesx_{i}. we introduce quasi tensor product space anddiscuss about the relation between tensor product space and thisnew space which we denote it by x boxtimesy. we investigate best approximation in direct sum of lattice normed spaces by elements which are not necessarily downwardor upward a...

We first show that a bounded linear operator $ T $ on a real Banach space $ E $ is quasicompact (Riesz, respectively) if and only if $T': E_{mathbb{C}}longrightarrow E_{mathbb{C}}$ is quasicompact  (Riesz, respectively), where the complex Banach space $E_{mathbb{C}}$ is a suitable complexification of $E$ and $T'$ is the complex linear operator on $E_{mathbb{C}}$ associated with $T$. Next, we pr...

We use functional calculus methods to investigate qualitative properties of C0semigroups that are preserved by time-discretization methods. Preservation of positivity, concavity and other qualitative shape properties which can be described via positivity are treated in a Banach lattice framework. Preservation of contractivity (or norm-bound) of the semigroup is investigated in the Banach space ...

We prove that a homogeneous Banach space B on the unit circle T can be embedded as a closed subspace of a dual space ΞB contained in the space of bounded Borel measures on T in such a way that the map B → ΞB defines a bijective correspondence between the class of homogeneous Banach spaces on T and the class of prehomogeneous Banach spaces on T. We apply our results to show that the algebra of a...

We construct a quasi-Banach space which cannot be given an equivalent plurisubharmonic quasi-norm, but such that it has a quotient by a onedimensional space which is a Banach space. We then use this example to construct a compact convex set in a quasi-Banach space which cannot be atfinely embedded into the space L0 of all measurable functions.

Let X be a completely regular topological space, B(X) the Banach space of real-valued bounded continuous functions on X, with the usual norm ||&|| =supa?£x|&(#)| • A subset GCB(X) is called completely regular (c.r.) over X if given any closed subset KQ.X and point XoÇzX — K, there exists a ô £ G such that &(#o) = |NI a n ( i sup^^is: \b(x)\ <||&||. A topological space X is completely regular in...

Every Banach space X with the Banach-Saks property is reflexive, but the converse is not true (see [4, 5]). Kakutani [6] proved that any uniformly convex Banach space X has the Banach-Saks property. Moreover, he also proved that if X is a reflexive Banach space and θ ∈ (0, 2) such that for every sequence (xn) in S(X) weakly convergent to zero, there exist n1, n2 ∈ N satisfying the Banach-Saks p...

The real geometric properties of spaces of polynomials are discussed in [1, 6]. In particular, it is shown that the symmetric injective tensor product space ⊗̂n,s,εE is not strictly convex if E is a Banach space of dimE ≥ 2 and if n ≥ 2 holds. Let E be a Banach space over a real or complex filed and E is denoted as the Banach dual of E. An element x in the unit sphere SE is called a (real) extre...