Banach Journal of Mathematical Analysis

On linear functional equations and completeness of normed spaces

Ajda Fosner, Roman Ger, Attila Gilanyi, and Mohammad Sal Moslehian

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Abstract

The aim of this note is to give a type of characterization of Banach spaces in terms of the stability of functional equations. More precisely, we prove that a normed space $X$ is complete if there exists a functional equation of the type $$\sum_{i=1}^{n}a_if(\varphi_i(x_1,\ldots,x_k))=0 \qquad(x_1,\ldots,x_k\in D)$$ with given real numbers $a_1,\ldots,a_n$, given mappings $\varphi_1\ldots,\varphi_n\colon D^k\to D$ and unknown function $f\colon D\to X$, which has a Hyers--Ulam stability property on an infinite subset $D$ of the integers.

Article information

Source
Banach J. Math. Anal., Volume 7, Number 1 (2013), 196-200.

Dates
First available in Project Euclid: 22 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1358864559

Digital Object Identifier
doi:10.15352/bjma/1358864559

Mathematical Reviews number (MathSciNet)
MR3004277

Zentralblatt MATH identifier
1282.46021

Subjects
Primary: 46B99: None of the above, but in this section
Secondary: 39B82: Stability, separation, extension, and related topics [See also 46A22]

Keywords
Hyers-Ulam stability normed space completeness Banach space

Citation

Fosner, Ajda; Ger, Roman; Gilanyi, Attila; Sal Moslehian, Mohammad. On linear functional equations and completeness of normed spaces. Banach J. Math. Anal. 7 (2013), no. 1, 196--200. doi:10.15352/bjma/1358864559. https://projecteuclid.org/euclid.bjma/1358864559


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