Banach空间的若干几何性质及线性算子度量广义逆的研究

On the basis of functional analysis, we find some new convex (smooth) Banach spaces and give the characteristic inequalities of these convex (smooth) Banach spaces or some known convex (smooth) Banach spaces, and begin to develop the duality theory of convexity and smoothness. We further investigate the several dentabilities which were introduced in our early study and other new dentability, and explain the relationship between these dentabilities and Radon–Nikodym property (or Krein–Milman property). We investigate the ball–covering property in Banach spaces and dicuss the relationship between the ball–covering property and the convex analysis in infinite dimensional spaces. We investigate the generalized inverse theory of Banach spaces and give the norm (error) estimate of stable perturbation for set–valued metric generalized inverse of bounded linear operators. Some relevant results about set–valued metric generalized inverse are extend to quasi–linear generalized inverse. This work will promoted the development of geometric theory of Banach spaces and operator spaces, and have also with good deal of applications, such as in optimization theory, approximation theory, geometric theiry of convex boby, control theory as well as fixed point theory and so on.

以泛函分析为基础， 基于Banach空间理论，发现具有某种新凸性(新光滑性)的Banach空间，给出这些凸性(或光滑性)空间以及已有的某些凸性(或光滑性)空间的特征不等式，开展凸性和光滑性之间的对偶理论的研究。继续研究我们在前期研究中引入的几种可凹性以及新可凹性，揭示这些可凹性与Radon–Nikodym性质(或Krein–Milman性质)的关系。研究Banach空间的球覆盖性质，并讨论球覆盖性质和无穷维空间凸分析之间的关系。研究Banach空间广义逆理论，给出有界线性算子的集值度量广义逆的稳定扰动的范数估计和误差估计，将集值度量广义逆的相关结果推广至拟线性广义逆。该研究中将要得到的结果对Banach空间理论和算子理论的发展有重大的理论价值和学术意义，在不动点理论、优化理论、逼近理论，凸体的几何理论和控制理论等领域有广泛的应用前景。

以泛函分析为基础， 基于Banach空间理论，发现具有某种新凸性(新光滑性)的Banach空间，得到了这些凸性(或光滑性)空间的特征刻画以及这些空间的一些几何性质，建立了新的凸性空间和某些凸性空间的特征不等式，开展了凸性和光滑性之间的对偶理论的研究. 继续研究了已有的某些可凹性等几何性质，引入了几种新的可凹性，并以它们为工具刻画了三类已知的光滑性空间. 研究了Banach空间的球覆盖性质，并讨论了球覆盖性质和无穷维空间凸分析之间的关系. 研究了Banach空间广义逆理论，给出了有界线性算子的集值度量广义逆的某些重要结果. 拓宽了研究范围，研究了Hamilton算子的谱问题以及时滞微分系统的控制理论. 该研究中得到的结果对Banach空间理论和算子理论的发展有重大的理论价值和学术意义，在不动点理论、优化理论、逼近理论，凸体的几何理论和控制理论等领域有广泛的应用前景.