Product spaces generated by bilinear maps and duality

Continuity of bilinear maps on direct sums of topological vector spaces

We prove a criterion for continuity of bilinear maps on countable direct sums of topological vector spaces. As a first application, we get a new proof for the fact (due to Hirai et al. 2001) that the map taking a pair of test functions on R^n to their convolution is continuous. The criterion also allows an open problem by K.-H. Neeb to be solved: If E is a locally convex space, regard the tensor algebra T(E) as the locally convex direct sum of the projective tensor powers T^j(E) of E. We show that T(E) is a topological algebra if and only if every sequence of continuous seminorms on E has an u…

Upper bounds for continuous seminorms and special properties of bilinear maps

If E is a locally convex topological vector space, let P(E) be the pre-ordered set of all continuous seminorms on E. We study, on the one hand, for g an infinite cardinal those locally convex spaces E which have the g-neighbourhood property in the sense of E. Jorda, i.e., spaces in which all sets M of continuous seminorms of cardinality up to g have an upper bound in P(E). On the other hand, we study bilinear maps b from a product of locally convex spaces E_1 and E_2 to a locally convex space F, which admit "product estimates" in the sense that for all p_{i,j} in P(F), i,j=1,2,..., there exist…

Applications of hypocontinuous bilinear maps in infinite-dimensional differential calculus

Paradigms of bilinear maps f between locally convex spaces (like evaluation or composition) are not continuous, but merely hypocontinuous. We describe situations where, nonetheless, compositions of f with Keller C^n_c-maps (on suitable domains) are C^n_c. Our main applications concern holomorphic families of operators, and the foundations of locally convex Poisson vector spaces.

在紧元处具有乘积性质的 C*-代数上的双线性映射

Bilinear maps on C$^*$-algebras that have product property at a compact element

本文研究 C*-代数上在紧元处具有乘积性质的有界双线性映射，分别对紧 C*-代数和具有非平凡原子部分的 von Neumann 代数给出何时由该点乘积确定的判据，并应用于刻画固定点处类似同态与类似导子的映射。

Duality for ideals of Lipschitz maps

We develop a systematic approach to the study of duality for ideals of Lipschitz maps from a metric space to a Banach space, inspired by the classical theory that relates ideals of operators and tensor norms for Banach spaces, by using the Lipschitz tensor products previously introduced by the same authors. We first study spaces of Lipschitz maps, from a metric space to a dual Banach space, that can be represented canonically as the dual of a Lipschitz tensor product endowed with a Lipschitz cross-norm. We show that several known examples of ideals of Lipschitz maps (Lipschitz maps, Lipschitz …

Quasilinear duality and inversion in Banach spaces

We present a unified approach to the processes of inversion and duality for quasilinear and $1$-quasilinear maps; in particular, for centralizers and differentials generated by interpolation methods.

Yosida Duality

In this note we prove Yosida duality --- that is: the category of compact Hausdorff spaces with continuous maps is dually equivalent to the category of uniformly complete Archimedean Riesz spaces with distinguished units and unit-preserving Riesz homomorphisms between them.

极值长度与对偶性

Extremal length and duality

本文集中于极值长度理论的对偶性性质及其在拟共形分析中的应用，展示了该理论的灵活性，并给出了度量曲面一致化、$n$ 维空间间的拟共形映射及多重连通平面区域间共形映射三种背景下的最新进展。