On the Mazur--Ulam theorem in fuzzy n--normed strictly convex spaces

In this paper, we generalize the Mazur--Ulam theorem in the fuzzy real n-normed strictly convex spaces.Comment: 7 page

Decompositions in direct sum of seminormed vector spaces and Mazur-Ulam theorem

It was proved by S. Mazur and S. Ulam in 1932 that every isometric surjection between normed real vector spaces is affine. We generalize the Mazur--Ulam theorem and find necessary and sufficient conditions under which distance-preserving mappings between seminormed real vector spaces are linear.Comment: 7 page

The Mazur-Ulam property for a Banach space which satisfies a separation condition (Research on preserver problems on Banach algebras and related topics)

After some preparations in section 1, we recall the three well known concepts: the Choquet boundary, the Šilov boundary, and the strong boundary points in section 2. We need to define them by avoiding the confusion which appears because of the variety of names of these concepts; they sometimes differs from authors to authors. We describe the relationship between the three concepts emphasizing the case where a function space strongly separates the points in the underlying space. We study C-rich spaces, lush spaces, and extremely C-regular spaces concerning with the Mazur-Ulam property in section 3. We show that a uniform algebra and the uniform closure of the real part of a uniform algebra with the supremum norm are C-rich spaces, hence lush spaces. We prove that a uniformly closed subalgebra of the algebra of all complex-valued continuous functions on a locally compact Hausdorff space which vanish at infinity is extremely C-regular provided that it separates the points of the underlying space and has no common zeros. We exhibit a space of harmonic functions which has the Mazur- Ulam property (Corollary 3.8). The main concern in sections 4 through 6 is the Mazur-Ulam property. We exhibit a sufficient condition on a Banach space which has the Mazur-Ulam property and the complex Mazur-Ulam property (Propositions 4.11 and 4.12). In sections 5 and 6 we consider a Banach space with a separation condition (∗) (Definition 5.1). We prove that a real Banach space satisfying (∗) has the Mazur-Ulam property (Theorem 6.1), and a complex Banach space satisfying (∗) has the complex Mazur-Ulam property (Theorem 6.3). Applying these theorems and the results in the previous sections we prove that an extremely C-regular real (resp. complex) linear subspace has the (resp. complex) Mazur-Ulam property (Corollary 6.2 (resp. 6.4)) in section 6. As a consequence we prove that any closed subalgebra of the algebra of all complex-valued continuous functions defined on a locally compact Hausdorff space has the complex Mazur-Ulam property (Corollary 6.5)

Perturbations of isometries between Banach spaces

We prove a very general theorem concerning the estimation of the expression $\|T(\frac{a+b}{2}) - \frac{Ta+Tb}{2}\|$ for different kinds of maps $T$ satisfying some general perurbated isometry condition. It can be seen as a quantitative generalization of the classical Mazur-Ulam theorem. The estimates improve the existing ones for bi-Lipschitz maps. As a consequence we also obtain a very simple proof of the result of Gevirtz which answers the Hyers-Ulam problem and we prove a non-linear generalization of the Banach-Stone theorem which improves the results of Jarosz and more recent results of Dutrieux and Kalton

Mazur-Ulam Theorem

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