拟Banach空间的几何性质和向量积

本项目从建立任意两个拓扑向量空间正张量积的表示的创新方法出发，利用拓扑向量空间正张量积的表示来研究拓扑向量空间的复凸性和Radon-Nikodym性质等各种重要性质的向量积的继承性问题。对拟Banach空间等拓扑向量空间的向量积的继承性进行深入的研究，提取分析出向量积可以继承的相关的几何性质，然后将讨论和研究这些几何性质在拟Banach格正张量积的继承性，并得到它们较好的刻画。建立一种全新的理论和方法来研究无限阶向量积的特征值和特征向量，进而研究拓扑向量空间和拟Banach格无限阶向量积理论，得到拟Banach格无限阶向量积的Kadec-Klee性质和Radon-Nikodym性质等的继承性刻画。此项工作将推进拓扑向量空间和拟Banach格理论研究方法的变革，并推动拓扑向量空间理论与拟Banach格的交叉领域和研究的深入发展。

This project starting from the establishment the representation of the positive tensor product of two topological vector spaces, by using the representation of the positive tensor product of topological vector spaces to study the inheritance of complex convexity, Radon-Nikodym property and other important properties in tensor product. The inheritance in tensor product of Quasi-Banach spaces will be researched in depth, we can extract the geometric properties in the tensor product those can be inherited, and then discuss and study the inheritance of these geometric properties in the tensor product of Quasi-Banach lattices, and attain their characterization. Establish a new theory and methods to study the eigenvalues and eigenvectors, and study the infinite order tensor product of topological vector spaces, then attain the characterization of inheritance about the Kadec-Klee property and the Radon-Nikodym property in the infinite order tensor product of Quasi-Banach lattices. The work will promote the change of the research methods in topological vector spaces and Quasi-Banach lattices, and to promote the development of the intersection areas of topological vector spaces and Quasi-Banach lattices.