# Download Isometries on Banach spaces: function spaces by Richard J. Fleming PDF

By Richard J. Fleming

Primary to the research of any mathematical constitution is an realizing of its symmetries. within the category of Banach areas, this leads evidently to a learn of isometries-the linear differences that look after distances. In his foundational treatise, Banach confirmed that each linear isometry at the area of constant capabilities on a compact metric house needs to rework a continual functionality x right into a non-stop functionality y pleasant y(t) = h(t)x(p(t)), the place p is a homeomorphism and |h| is identically one.Isometries on Banach areas: functionality areas is the 1st of 2 deliberate volumes that survey investigations of Banach-space isometries. This quantity emphasizes the characterization of isometries and specializes in constructing the kind of particular, canonical shape given above in quite a few settings. After an introductory dialogue of isometries more often than not, 4 chapters are dedicated to describing the isometries on classical functionality areas. the ultimate bankruptcy explores isometries on Banach algebras.This therapy offers a transparent account of traditionally vital effects, exposes the relevant equipment of assault, and contains a few effects which are more moderen and a few which are lesser identified. precise in its concentration, this ebook will end up precious for specialists in addition to novices within the box and in the event you easily are looking to acquaint themselves with this sector of Banach house idea.

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NOTES AND REMARKS 43 One type of generalization that has been popular is to weaken the assumptions about the topological spaces. Thus Jarosz and Pathak consider isometries on subspaces of C ( Q ) given other norms [I501 while Araujo and Jarosz [12] obtain the canonical description of isometries on metric spaces of unbounded continuous functions defined on noncompact topological spaces. Similarly, Bachir [19] has extended the Banach-Stone theorem to certain subspaces of the bounded continuous functions on a complete metric space.

This generalizes a result due to Holsztynski [135] and Lovblom [202]. Drewnowski's paper contains a number of interesting results about the question of linearity. There is also a good discussion of the Mazur-Ulam Theorem in Day [83]. 6 is due to Figiel [99] who proved the following theorem which he says was conjectured by Holsztynski and Lindenstrauss. 3. (Figiel) Let X and Y be two real Banach spaces and U an isometry (not necessarily linear) mapping X into Y such that U(0) = 0. If the linear span of U ( X ) is dense in Y , then there is a continuous linear operator F mapping Y into X such that the composition F o U is the identity on X .

4. A THEOREM O F VESENTINI 39 we conclude that (0,2) has no points in common with ch(N) which therefore equals ( - m , 0] U [2, m ) . (iv) Let M = {f E C[O, 11 : f ( 0 ) = f ( 1 ) = 0). : = [O, 11, and 2 r = [O, 1/21, In this case ch(M) = ( 0 , l ) and ch(N) = (0,112) where N = T(M). 4. A Theorem of Vesentini It is well known that the extreme points of the unit ball of C ( Q ) are those functions f such that If (s)l = 1 for all s E Q. For any subspace M of C ( Q ) , let r ( M ) denote the set of functions in M such that I f(s)l = 1 for all s E ch(M).