We give a characterization of Banach spaces X containing a subspace with a shrinking Markushevich basis {xγ , fγ}γ∈Γ. This gives a sufficient condition for X to have a renorming with Mazur’s intersection property. A biorthogonal system in a Banach space X is a subset {xγ , fγ}γ∈Γ ⊂ X×X such that fγ(xγ′) = δγγ′ for γ, γ ′ ∈ Γ. The biorthogonal system {xγ, fγ}γ∈Γ in X is called fundamental if X =...

In this paper we show that a result of Gross and Kuelbs, used to study Gaussian measures on Banach spaces, makes it possible to construct an adjoint for operators on separable Banach spaces. This result is used to extend well-known theorems of von Neumann and Lax. We also partially solve an open problem on the existence of a Markushevich basis with unit norm and prove that all closed densely de...

We show that the class of subspaces of c0(N) is stable under Lipschitz isomorphisms. The main corollary is that any Banach space which is Lipschitz-isomorphic to c0(N) is linearly isomorphic to c0(N). The proof relies in part on an isomorphic characterization of subspaces of c0(N) as separable spaces having an equivalent norm such that the weak-star and norm topologies quantitatively agree on t...

We construct an example of a real Banach space whose group of surjective isometries reduces to ± Id, but the group of surjective isometries of its dual contains the group of isometries of a separable infinite-dimensional Hilbert space as a subgroup. To do so, we present examples of extremely non-complex Banach spaces (i.e. spaces X such that ‖ Id+T ‖ = 1+‖T ‖ for every bounded linear operator T...

Let E be a Banach space with a separable dual. Zippin’s theorem asserts that E embeds in a Banach space E1 with a shrinking basis, and W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński have shown that E is a quotient of a Banach space E2 with a shrinking basis. These two results use the interpolation theorem established by W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński. Here, we ...

A well known argument of James yields that if a Banach space X contains `1 ’s uniformly, then X contains `1 ’s almost isometrically. In the first half of the paper we extend this idea to the ordinal `1-indices of Bourgain. In the second half we use our results to calculate the `1-index of certain Banach spaces. Furthermore we show that the `1-index of a separable Banach space not containing `1 ...

In any Banach space a monotone operator with a norm-to-weak upper semicontinuous multivalued selection on an open set D is singlevalued and norm-to-norm upper semicontinuous at the points of a dense G subset of D. Monotone operators | and especially a special case of them, subdiierentials of convex functions | play an important role in various parts of nonlinear analysis. One of the often inves...

In any Banach space a monotone operator with a norm-to-weak upper semicontinuous multivalued selection on an open set D is singlevalued and normto-norm upper semicontinuous at the points of a dense Gδ subset of D. Monotone operators — and especially a special case of them, subdifferentials of convex functions — play an important role in various parts of nonlinear analysis. One of the often inve...

Let X be a Banach space with closed unit ball B. Given k ∈ N, X is said to be k-β, repectively, (k + 1)-nearly uniformly convex ((k + 1)-NUC), if for every ε > 0, there exists δ, 0 < δ < 1, so that for every x ∈ B, and every ε-separated sequence (xn) ⊆ B, there are indices (ni) k i=1, respectively, (ni) k+1 i=1 , such that 1 k+1 ‖x + ∑k i=1 xni‖ ≤ 1 − δ, respectively, 1 k+1 ‖ ∑k+1 i=1 xni‖ ≤ 1−...

The paper investigates the variation of the spectrum of operators in infinite dimensional Banach spaces. Consider the space of bounded operators on a separable Banach space when equipped with the strong operator topology, and the Polish space of compact subsets of the closed unit disc of the complex plane when equipped with the Hausdorff topology. Then, it is shown that the unit spectrum functi...