Inner Product Groups and Riesz Representation Theorem

In this paper, we introduce an inner product on abelian groups and, after investigating the basic properties of the inner product, we first show that each inner product group is a torsion-free abelian normed group. We give examples of such groups and describe the norms induced by such inner products. Among other results, Hilbert groups, midconvex and orthogonal subgroups are presented, and a Riesz representation theorem on divisible Hilbert groups is proved.

Development and Validation of the German Test for (Highly) Intelligent Kids – T(H)INK

Background: Currently, no recently normed group intelligence test for primary school children is available in Germany to cover verbal, numerical, and figural reasoning abilities. Furthermore, no test for this age group is difficult enough to be able to differentiate especially at the upper end of the intelligence distribution. Aims: The Test for (Highly) Intelligent Kids – T(H)INK was developed to fill this gap. Methods: A total of 1,629 children from grades 1 to 3 of primary school took the 36-item test (one version per class level). Results: At all three class levels, the test proved sufficiently reliable (Cronbach’s α = .73–.80 for the full scale). Retest reliabilities after 1 year, assessed for a random subsample, ranged from .61 to .73. Factorial validity was established by exploratory and confirmatory factor analyses, with a g factor explaining 57.61–63.67% of the variance. Furthermore, convergent validity (correlations with a nonverbal intelligence test, teacher and parent ratings, grades, and the probability for the child to be nominated for a gifted and talented program) was satisfying, as was the predictive validity for average marks a year later. Discussion: Implications of the findings and ideas for future research and development of the test are discussed.