On Inner Products in Linear, Metric Spaces

Borzellino, Joseph E.; Sherman, Morgan

doi:10.2307/1968653

Cited by 303 publications

(170 citation statements)

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“…The reason for this is that M. Fréchet [12] used it to characterize the inner product spaces in a similar way as Jordan and von Neumann [15] did using the parallelogram law. Namely, he proved that a normed space (X, · ) is an inner product space if and only if, for all x, y, z ∈ X,…”

Section: Renata Malejkimentioning

confidence: 99%

Stability of a generalization of the Fréchet functional equation

Malejki¹

2015

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

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Section: Renata Malejkimentioning

confidence: 99%

Stability of a generalization of the Fréchet functional equation

Malejki¹

2015

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

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“…For n = 0 condition (35) follows directly from (20). Assume that (35) holds for some n ∈ N 0 and all (x, y, z) ∈ X 3 .…”

Section: Now We Will Show That Condition (H2) Of Theorem 1 Is Satisfimentioning

confidence: 99%

“…(2), since its assumptions exclude the case where (18), and β 0 , β ∈ [0, 1), where β 0 and β are defined as in Theorem 13. Let f : X → Y be a function satisfying condition (20). Then there exists a unique additive solution a :…”

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confidence: 99%

On the generalized Fréchet functional equation with constant coefficients and its stability

2018

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Abstract. We study a generalization of the Fréchet functional equation, stemming from a characterization of inner product spaces. We show, in particular, that under some weak additional assumptions each solution of such an equation is additive. We also obtain a theorem on the Ulam type stability of the equation. In its proof we use a fixed point result to show the existence of an exact solution of the equation that is close to a given approximate solution.Mathematics Subject Classification. 39B52, 39B82, 47H10.

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“…for all x 1 , ..., x n ∈ X (see also [2,19]). During the last three decades a number of papers and research monographs have been published on various generalizations and applications of the generalized Hyers-Ulam stability to a number of functional equations and functions (see [5]- [14], [17,18,21,22] and [26]- [29]).…”

Section: Then There Exists a Unique Additive Functionmentioning

confidence: 99%