Sets of weak sequential continuity for polynomials
Abstract Let P : E → K be an N-homogeneous polynomial, where E is a Banach space over K = R or C . We study properties of the set Cp = {x ∈ E : P is weakly sequentially continuous at x}.
Block diagonal polynomials
In this paper we introduce and study a certain class of polynomials in spaces with unconditional finite dimensional decomposition. Some applications to the existence of copies of `∞ in spaces of polynomials and to the stabilization of polynomial algebras are given. Introduction Motivated by several problems related to the structure of spaces of polynomials, we introduce and study the class of block diagonal polynomials in the context of spaces with unconditional finite dimensional decomposition. This kind of polynomials will be useful to attack two different problems. On one hand, for many examples in the literature it is noted that the space of homogeneous polynomials contains an isomorphic copy of `∞ (conditions to ensure this fact can be found in [12, 13, 9, 14, 10, 16]). Also, it is proved in [9] that there exists a dichotomy between the fact that all polynomials are weakly sequentially continuous and the existence of isomorphic copies of `∞ in the space of polynomials. Here we are concerned with the “position” of such isomorphic copies of `∞ and the relationship between weak sequential continuity of polynomials and block diagonal polynomials. The other problem is to study the stabilization of polynomial algebras [23]. In this direction, we describe the behavior of the chain of algebras generated by polynomials up to a certain degree in the particular case of Banach spaces with unconditional finite dimensional decomposition. In Section 1, we introduce the class of block diagonal polynomials and establish some properties of these polynomials. We provide a technique to construct, from a fixed polynomial, a block diagonal polynomial which preserves some properties of the original one. Using this, we prove the equivalence between weak sequential continuity of polynomials and the same property for block diagonal polynomials. Moreover, if there exist polynomials which are not weakly sequentially continuous, this technique turns out to be useful to show how to embed an isomorphic copy of `∞ in a certain space of block diagonal polynomials. The second section is totally devoted to studying the simplest class of block diagonal polynomials in a space with unconditional basis, namely, diagonal polynomials. In the case of real Banach spaces with 1-unconditional basis, considered Received by the editors July 24, 1998 and, in revised form, July 22, 1999. 2000 Mathematics Subject Classification. Primary 46G20; Secondary 46Bxx.
Imaging the in-plane magnetization in a Co microstructure by Fourier transform holography.
We report on experiments using Fourier transform holography to image the in-plane magnetization of a magnetic microstructure. Magnetic sensitivity is achieved via the x-ray magnetic circular dichroism effect by recording holograms in transmission at off-normal incidence. The reference beam is defined by a narrow hole milled at an inclined angle into the opaque mask. We present magnetic domain images of an in-plane magnetized cobalt element with a size of 2 μm × 2 μm× 20 nm. The domain pattern shows a multi-vortex state that deviates from the simple Landau ground state.