caa a22 4500 605508127 CHVBK 20210128100635.0 cr unu---uuuuu 210128e20150201xx s 000 0 eng 10.1007/s00010-014-0299-x doi (NATIONALLICENCE)springer-10.1007/s00010-014-0299-x Ger Roman Institute of Mathematics, Silesian University, ul. Bankowa 14, PL-40-007, Katowice, Poland aut Fischer-Muszély functional equation almost everywhere [Elektronische Daten] [Roman Ger] We show that, under suitable assumptions, a function f from a group (G, +) into a real or complex inner product space $${(H, (\cdot \vert \cdot))}$$ ( H , ( · | · ) ) , satisfying the Fischer-Muszély functional equation $$\parallel f(x + y)\parallel\, \, =\, \parallel f(x) + f(y)\parallel$$ ‖ f ( x + y ) ‖ = ‖ f ( x ) + f ( y ) ‖ for all pairs (x, y) off a sufficiently small (negligible) subset of G 2 has to be almost everywhere equal to an additive map from G into H, i.e. the set $${\{x \in G: f(x) \neq a(x)\}}$$ { x ∈ G : f ( x ) ≠ a ( x ) } is small (negligible) in G. Small sets in G and G 2 are defined in an axiomatic way. Several corollaries illustrating some consequences of this result are presented. The Author(s), 2014 Fischer-Muszély equation nationallicence p.l.i. ideal nationallicence conjugate ideals nationallicence equality (equation) valid $${\mathcal{J}}$$ J -almost everywhere ( $${\Omega(\mathcal{J})}$$ Ω ( J ) -almost everywhere) nationallicence Aequationes mathematicae Springer Basel 89/1(2015-02-01), 207-214 0001-9054 89:1<207 2015 89 10 https://doi.org/10.1007/s00010-014-0299-x text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00010-014-0299-x text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Ger Roman Institute of Mathematics, Silesian University, ul. Bankowa 14, PL-40-007, Katowice, Poland aut NATIONALLICENCE 773 0- Aequationes mathematicae Springer Basel 89/1(2015-02-01), 207-214 0001-9054 89:1<207 2015 89 10