American Institute of Mathematical Sciences

October  2007, 8(3): 557-567. doi: 10.3934/dcdsb.2007.8.557

On the generalized pythagorean parameters and the applications in Banach spaces

 1 Department of Mathematics, Community College of Philadelphia, Philadelphia, PA 19130-3991, United States

Received  August 2006 Revised  January 2007 Published  July 2007

Let X be a Normed space and $S(X) = \{x \in X : \|\|x\|\| = 1\}$ be the unit sphere of X. Following the previous results for the Pythagorean approach in Banach spaces [5], [6], the generalized parameters $E_{\xi, \eta}(X)=$sup${\alpha_{\eta}(\xi x): x in S(X)\}$, $e_{\xi, \eta}(X)=$inf$\{\alpha_{\eta}(\xi x): x \in S(X)\}$, $F_{\xi, \eta}(X)=$sup${\beta_{\eta}(\xi x): x \in S(X)\}$, and $f_{\xi, \eta}(X)=$inf${\beta_{\eta}(\xi x): x \in S(X)\}$, where $\alpha_{\eta}(\xi x) =$sup${||\xi x + \eta y ||^2+ ||\xi x - \eta y ||^{2}: y \in S(X)\}$, $\beta_{\eta}(\xi x) =$inf${\|\|\xi x + \eta y ||^{2}+ ||\xi x - \eta y ||^{2}: y \in S(X)\}$ and $\xi, \eta > 0$ are defined and studied. The values of these parameters of some classical normed spaces are estimated and the relationship of these parameters with other geometric properties are investigated, and some existing results are extended also.
Citation: Ji Gao. On the generalized pythagorean parameters and the applications in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 557-567. doi: 10.3934/dcdsb.2007.8.557
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