The Mazur-Ulam Property On Vector-valued Fuction Space

Let X and Y be real normed spaces.If for any x,y ∈X,‖V(x)-V(y)‖=‖x-y‖,then it is called isometric operator.Tingley raised the famous Tingley’s problem in 1987:let X and Y be normed spaces with unit spheres SX and SY respectively.Suppose that V:SX→SY is a surjective isometry.Is there a linear isometry V:X→Y such that V|SX=V?The classical Mazur-Ulam theorem shows that every surjective isometry between two normed spaces is affine.The Mazur-Ulam property(MUP)were proposed by Cheng Lixin and Dong Yunbai in 2011:Let X be a real Banach space.For every Banach space Y,every surjective isometry between the unit spheres of X and Y can be extended to a real linear isometry on the whole space.Then X is said to have the Mazur-Ulam property.J.D.Hardtke put forward that:Every Banach space with property(**)has the Mazur-Ulam property.A study is made on whether the vector space C(K,X)、L∞(μ,X)and L1(μ,X)has the same property if X has property(**).On the contrary,is it true?In the first chapter,we introduce the current research situation of Mazur-Ulam property and Tingley’s problem,and introduce the definition of Generalized-lush(GL)space,vector valued space and the research results of several classical spaces,which leads to the problems that need to be proved later.In the second chapter,we study C(K,X)and L∞(μ,X)spaces,introduce some stability property in these two spaces,and draw a conclusion through the definition of property(**):if X has the property(**),then C(K,X)and L∞(μ,X)space also have the Mazur-Ulam property and vice versa.In chapter 3,we first give the definition of space L1(μ,X)and its stability,then we describe some lemmas,and we prove that the property(**)is stable under c0-、l1-and l∞-direct sum,and finally we prove by the application of these lemmas and the definition of property(**)that a Banach space X has property(**)if and only if space L1(μ,X)has property(**).Similarly,if X has property(**),then space L1(μ,X)has the Mazur-Ulam property.