Nettet1 Answer. This is true for real vector spaces by the Mazur-Ulam theorem which states that a surjective distance-preserving linear map of one real normed space onto another is … Nettet6. des. 2012 · The purpose of this book is to present the main structure theorems in the isometric theory of classical Banach spaces. Elements of general topology, measure …

Invertibility Methods in Abstract Operator - Studocu

NettetWe prove that every surjective isometry between unit spheres of L∞(Σ,Ω,μ) L ∞ ( Σ, Ω, μ) and a Banach space F F can be linearly and isometrically extended to the whole space, which means that if the unit sphere of a Banach space F F is isometric to the unit sphere of L∞(Σ,Ω,μ) L ∞ ( Σ, Ω, μ), then F F is linearly isometric to L∞(Σ,Ω,μ) L ∞ ( Σ, … Nettet1. apr. 2000 · In the last section the problem of the existence of order linearly isometric copies of \(L_p(\nu )\) with \(0 thomas gigi band

On the Extension of Isometries between Unit Spheres of

Nettet10.4. The Unitary Group, Unitary Matrices 299 Remarks: (i) In the Euclidean case, we proved that the assumption f(v)−f(u) = v −u for all u,v ∈ E and f(0) = 0 (2 ) implies … Nettet10.4. The Unitary Group, Unitary Matrices 299 Remarks: (i) In the Euclidean case, we proved that the assumption f(v)−f(u) = v −u for all u,v ∈ E and f(0) = 0 (2 ) implies (3). For this we used the polarization identity NettetWe obtain that if the set sm. S 1 ( E) of all smooth points of the unit sphere S 1 ( E) is dense in S 1 ( E ), then under some condition, every surjective isometry V 0 from S 1 ( E) onto S 1 ( C (Ω)) can be extended to be a real linearly isometric map V of E onto C (Ω). From this result we also obtain some corollaries. thomas gilbert behavioral engineering model