Basic and quasibasic subspaces of adjoint Banach spaces

Translated from Matematicheskie Zametki, Vol. 47, No. 6, pp. 85–90, June, 1990.     

Banach spaces without minimal subspaces

We prove three new dichotomies for Banach spaces à la W.T. Gowers' dichotomies. The three dichotomies characterise respectively the spaces having no minimal subspaces, having no subsequentially minimal basic sequences, and having no subspaces crudely finitely representable in all of their subspaces. We subsequently use these results to make progress on Gowers' program of classifying Banach spaces by finding characteristic spaces present in every space. Also, the results are used to embed any partial order of size ℵ₁ into the subspaces of any space without a minimal subspace ordered by isomorphic embeddability.     

On pseudo-complemented subspaces of Banach spaces

As natural generalizations of complemented subspaces, we introduce pseudo-complemented and strongly pseudo-complemented subspaces of Banach spaces, and we study for such subspaces some of the problems analogous to the classical problems on complemented and quasi-complemented subspaces.     

Quasireflexive locally convex spaces without Banach subspaces

Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 44, pp. 78–84, 1985.     

On subspaces of Banach spaces without quasicomplements

It is proved that for uncountableΓ,c ₀(Γ) has no quasicomplement inm(Γ). Research supported by the National Science Foundation, U.S.A.     

On finite dimensional subspaces of Banach spaces

The common Banach spaces are investigated with respect to some properties of their finite dimensional subspaces. The contribution to the paper of the second author is part of his Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor A. Dvoretzky. The authors wish to thank Professor A. Dvoretzky for the interest he showed in the paper and for his helpful advice.     

Projections onto Hilbertian subspaces of Banach spaces

In this paper we obtain new estimates for the relative projection constants of subspaces of a Banach spaceY in terms of geometrical properties ofY. Our method gives thatK-convex spaces are locally π-Euclidean. We also get a version of Maurey’s extension theorem for spaces of typep<2.     

Independent sequences in Banach spaces

In every ∞-dimensional separable Banach spaceX there is a fundamental sequence such that no subsequence of it, which is fundamental inX, is independent (“{x _n} is fundamental inX” meansX=span {x _n}).     

Adjoint subspaces in Banach spaces,with applications to ordinary differential subspaces

Summary Given two subspaces A₀ ⊂ A₁ ⊂ W=X ⊕ Y, where X, Y are Banach spaces, we show how to characterize, in terms of generalized boundary conditions, those adjoint pairs A, A* satisfying A₀ ⊂ A ⊂ A₁, A ₁ ^* ⊂ A∗ ⊂ A ₀ ^* ⊂ W⁺=Y* ⊕ X*, where X*, Y* are the conjugate spaces of X, Y, respectively. The characterizations of selfadjoint (normal) subspace extensions of symmetric (formally normal) subspaces appear as special cases when Y=X*. These results are then applied to ordinary differential subspaces in W=L^q(ι) ⊕ L^r(ι), 1≦q, r≦∞, where τ is a real interval, and in W=C( ) ⊕ C( ), where is a compact interval. Entrata in Redazione il 21 febbraio 1977. The work of EarlA. Coddington was supported in part by the National Science Foundation under NSF Grant No. MCS-76-05855.     

On orthocomplemented subspaces in p-adic Banach spaces

For linear subspaces of finite-dimensional normed spaces over K, where K is a non-Archimedean complete valued field which is not spherically complete, we study orthocomplementation as related to strictness and the Hahn-Banach property. We prove that there exist finite-dimensional normed spaces which possess non-orthocomplemented, strict HB-subspaces.     

Banach spaces of typep have arbitrarily distortable subspaces

It is shown that if a Banach space has bounded distortions then it contains an unconditional basic sequence. It follows that Banach spaces of typep > 1 contain arbitrarily distortable subspaces. Furthermore, hereditarily indecomposable Banach spaces are themselves arbitrarily distortable.     

Strongly embedded subspaces of p-convex Banach function spaces

Let $X(\mu )$ be a p-convex ( $1\le p<\infty $ ) order continuous Banach function space over a positive finite measure  $\mu $ . We characterize the subspaces of  $X(\mu )$ which can be found simultaneously in  $X(\mu )$ and a suitable $L^1(\eta )$ space, where $\eta $ is a positive finite measure related to the representation of  $X(\mu )$ as an $L^p(m)$ space of a vector measure  $m$ . We provide in this way new tools to analyze the strict singularity of the inclusion of  $X(\mu )$ in such an $L^1$ space. No rearrangement invariant type restrictions on  $X(\mu )$ are required.     

On sum and quotient of quasi-Chebyshev subspaces in Banach spaces

It will be determined under what conditions types of proximinality are transmitted to and from quotient spaces.In the final section,by many examples we show that types of proximinality of subspaces in Banach spaces can not be preserved by equivalent norms.     

Onc 0 sequences in Banach spaces

A Banach space has property (S) if every normalized weakly null sequence contains a subsequences equivalent to the unit vector basis ofc ₀. We show that the equivalence constant can be chosen “uniformly”, i.e., independent of the choice of the normalized weakly null sequence. Furthermore we show that a Banach space with property (S) has property (u). This solves in the negative the conjecture that a separable Banach space with property (u) not containingl ₁ has a separable dual. This is part of this author's Ph.D. dissertation prepared at The University of Texas at Austin under the supervision of H. P. Rosenthal.     

Minimal subspaces and isomorphically homogeneous sequences in a Banach space

If a Banach space is saturated with subspaces with a Schauder basis, which embed into the linear span of any subsequence of their basis, then it contains a minimal subspace. It follows that any Banach space is either ergodic or contains a minimal subspace.     

Diagonalization of operators and complementation of subspaces of Banach spaces

We prove a theorem on the decomposition of the algebra of bounded operators into a direct sum of a subalgebra of diagonal operators and a subspace of nondiagonal operators.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 867–873, July, 1990.