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The first paper on characterization of inner product spaces was: P. Jordan and J. Von Neumann, On inner products in linear, metric spaces, Ann. of Math. (2) 36 (1935), no. 3, 719–723. There is a book on the subject: Vasile Ion Istrăţescu. Inner Product Structures: Theory and Applications. Springer, 1987. Especially chapter 4.


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Metric space $(X,\rho)$ satisfying Ptolemy inequality $\rho(a,b)\rho(c,d)+\rho(b,c)\rho(a,d)\geq \rho(a,c)\rho(b,d)$ is called ptolemaic space. A normed ptolemaic space must be inner product space. Reference: I.J. Schoenberg, A remark on M. M. Day’s characterization of innerproduct spaces and a conjecture of L. M. Blumenthal. Proc. Am. Math. Soc. 3, 961–964 ...


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This question is closely related to the so-called "condition number" of the B-spline basis. Basically, for a spline $f$ of some degree $p$ with a coefficient vector $c=(c_i)$, you generally have for any $q \in [1,\infty]$ that $$ A_{p,q} \|c\|_{\ell_q} \le \| f \|_{L_q} \le B_{p,q} \|c\|_{\ell_q}, $$ and the smallest possible ratio $\kappa_{p,q} = ...



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