This paper concerns a family of weak parallelogram laws for Banach spaces. It is shown that the familiar Lebesgue spaces satisfy a range of these inequalities. Connections are made to basic geometric ideas, such as smoothness, convexity, and Pythagorean-type theorems. The results are applied to the linear prediction of random processes spanning a Banach space. In particular, the weak parallelogram laws furnish coefficient growth estimates, Baxter-type inequalities, and criteria for regularity.
Copyright © 2015 Springer Netherlands
The definitive version is available at: http://link.springer.com/article/10.1007/s10998-014-0078-4
Ross, William T., and R. Cheng. "Weak Parallelogram Laws on Banach Spaces and Applications to Prediction." Periodica Mathematica Hungarica 71, no. 1 (2015): 45-58. doi:10.1007/s10998-014-0078-4.