Stability of Surjectivity

Abstract

We investigate the stability of surjective functions in Banach spaces. As a corollary to the main theorem we obtain a new stability result for isometries. Academic Press

Nonsurjective nearisometries of Banach spaces

“…Up to now the best known approximation estimates were 12ε + 5δ obtained by Dilworth [3, Prop. 2] and 2ε + 35δ obtained by J. Tabor [11]. The statement of our main result has also been known to be true under the additional condition that the set of points in X at which the norm is Fréchet differentiable is dense [3, Th.…”

A universal theorem for stability of ε-isometries of Banach spaces

“…The study of properties of non-surjective ε-isometries has been active since 90's of the last century (see, for instance, [1,3,[9][10][11][12][13]25,[27][28][29][30][31]). The question, if every standard ε-isometry f : X → Y admits a linear quasi-left inverse, that is, if there exists T ∈ B(Y f , X) so that T f − Id is uniformly bounded on X seems to be very natural.…”

Weak stability of 𝜺-isometry Mapping on Real Banach Spaces

“…Dilworth [6] showed that for every 𝛿surjective 𝜀-isometry mapping : f X Y → with f (0) = 0, where X and Y are Banach spaces, there exists an isometric mapping : U X Y → such that ( ) ( ) 12 5 f x U x  −  + . Then Tabor [23] changed this value to 2 35…”