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Using the notion of semi-inner product in normed spaces, in this paper we provide some new characterizations of inner product spaces. We answer a question posed by Dragomir, whether the property (N) is characteristic for inner product spaces. We show that the space X is of (N)-type if and only if the norm in X comes from an inner product. Mathematics subject classification (2010): 46C15, 46C50, 46B20, 39B62.

Characterizations of inner product spaces by inequalities involving semi-inner product