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On orthogonally additive injections and surjections

Baron K.

Commentationes Mathematicae

2015

Vol 55, No. 2

157--162

Let E be a real inner product space of dimension at least 2 and V a linear topological Hausdorff space. If card E≤card V, then the set of all orthogonally additive injections mapping E into V is dense in the space of all orthogonally additive functions from E into V with the Tychonoff topology. If cardV≤card E, then the set of all orthogonally additive surjections mapping E into V is dense in the space of all orthogonally additive functions from E into V with the Tychonoff topology.

On some consequences of the functional generalization of the parallelogram identity

Mączyński M. M.

Demonstratio Mathematica

2014

Vol. 47, nr 2

493--506

The aim of this paper is to unify the partial results, which up to now, have been dispersed in various publications in order to show the importance of the functional form of parallelogram identity in mathematics and physics. We study vector spaces admitting a real non-negative functional which satisfies an identity analogous to the parallelogram identity in normed vector spaces. We show that this generalized parallelogram identity also implies an equality analogous to the Cauchy–Schwarz inequality. We study the consequences of this identity in real and complex vector spaces, in generalized Riesz spaces and in abelian groups. We give a physical interpretation to these results. For vector spaces of observables and states, we show that the parallelogram identity implies an inequality analogous to Heisenberg’s uncertainty principle (HUP), and we show that we can obtain the standard structure of quantum mechanics from the parallelogram identity, without assuming from the beginning the HUP. The role of complex numbers in quantum mechanics is discussed.

On orthogonally additive functions with orthogonally additive second iterate

Baron K.

Commentationes Mathematicae

2013

Vol. 53, No. 2

301--304

Let E be a real inner product space of dimension at least 2. If f maps E onto E and both f and f∘f are orthogonally additive, then f is additive.

On inner product spaces - III

Kannapann P.

Bulletin of the Polish Academy of Sciences. Mathematics

1998

Vol. 46, no 1

1-4

When one deals with normed linear space (n.l.s.), the natural question arises when a n.l.s. is an inner product space (i.p.s.)? What further conditions the norm has to satisfy so that the n.l.s. an inner product space? Numerous charakterizations are known [2, 1, 2, 4, 5, 6, 7]. In this paper we study i.p.s. from functional equations point of view and consider three functional equations (ME), (14) and (15) which are generalizations of (LE) found in [6].