Abstract
In this paper, Let X, Y be two real Banach spaces and ε ≥ 0. A mapping f: X → Y is said to be a standard ε-isometry provided f(0) = 0 and
for all x, y ∈ X. If ε = 0, then it is simply called a standard isometry. We prove a sufficient and necessary condition for which {f(xn)}n≥1 is a basic sequence of Y equivalent to {xn}n≥1 whenever {xn}n≥1 is a basic sequence in X and f: X → Y is a nonlinear standard isometry. As a corollary we obtain the stability of basic sequences under the perturbation by nonlinear and non-surjective standard ε-isometries.
Similar content being viewed by others
References
F. Albiac and N. J. Kalton, Topics in Banach space theory, Graduate Texts in Mathematics 233, Springer, New York (2006).
Y. Benyamini and J. Lindenstrauss, Geometric nonlinear functional analysis I, Amer. Math. Soc. Colloquium Publications, Providence, RI, 48 (2000).
L. Cheng, Q. Cheng, K. Tu and J. Zhang, A universal theorem for stability of ε-isometries of Banach spaces, J. Funct. Anal., 269 (2015), 199–214.
D. Dai and Y. Dong, Stability of Banach spaces via nonlinear "-isometry, J. Math. Anal. Appl., 414 (2014), 996–1005.
T. Figiel, On non-linear isometric embeddings of normed linear spaces, Bull. Acad. Polon. Sci. Math. Astro. Phys., 16 (1968), 185–188.
D. Hyers and S. Ulam, On approximate isometries. Bull. Amer. Math. Soc., 51 (1945), 288–292.
J. Lindenstrauss and M. Zippin, Banach spaces with a unique unconditional basis, J. Funct. Anal., 3 (1969), 115–125.
S. Mazur and S. Ulam, Sur les transformations isométriques déspaces vectoriels normés. C.R. Acad. Sci. Paris., 194 (1932), 946–948.
M. S. Moslehian and G. Sadeghi, A Mazur-Ulam theorem in non-Archimedean normed spaces, Nonlinear Analysis: TMA., 69 (2008), 3405–3408.
M. Omladič, P. Šemrl, On non linear perturbations of isometries, Math. Ann., 303 (1995), 617–628.
H. P. Rosenthal, A characterization of Banach spaces containing, Proc. Natl. Acad. Sci. U.S.A., 71 (1974), 2411–2413.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the Natural Science Foundation of China (Grant No. 11601264) and the Outstanding Youth Scientific Research Personnel Training Program of Fujian Province and the Research Foundation of Quanzhou Normal University (Grant No. 2016YYKJ12) and the High level Talents Innovation and Entrepreneurship Project of Quanzhou City, (Grant No. 2017Z032).
Rights and permissions
About this article
Cite this article
Dai, D. Stability of basic sequences via nonlinear ε-isometries. Indian J Pure Appl Math 49, 571–579 (2018). https://doi.org/10.1007/s13226-018-0286-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13226-018-0286-3