pre-Hilbert space

pre-Hilbert space

[prē′hil·bərt ‚spās]
(mathematics)
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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([2], Proposition 1.14.) If a normed real vector space (X, ||*||) is CAT([kappa]), for some [kappa] [member of] R, then it is a pre-Hilbert space.
But according to Theorem 2.1 E is a CAT(K), for some [kappa] [member of] R, if and only if (E, ||*||) is a pre-Hilbert space. In other words, our definition gives a new class of CAT(0) metric spaces provided (E, ||*||) is not a pre-Hilbert vector space.