The known concept of 2-normed real linear spaces is extended to 2-normed complex linear spaces. This extension is not trivial. A Hahn-Banach type extension theorem for complex linear 2-functionals is established and it is shown that it is not possible to get this result from the known Hahn-Banach type extension theorem for real linear 2-functionals using the Bohnenblust-Sobczyk technique directly as is done in the case of linear functionals. As an application of our extension theorem, a 2-norm version of the Ascoli-Mazur theorem on tangent functionals is established. Several examples and counter examples illustrate the results obtained in the paper.
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S.N. Lal, S. Bhattacharya, C. Sreedhar, Complex 2-Normed Linear Spaces and Extension of Linear 2-Functionals. Z. Anal. Anwend. 20 (2001), no. 1, pp. 35–53DOI 10.4171/ZAA/1003