Zeitschrift f¨ ur Analysis und ihre Anwendungen Journal for Analysis and its Applications Volume 24 (2005), No. 3, 649–656 Vector Valued Baire Functions H. R. Shatery and J. Zafarani Abstract. In this paper, we study some properties of the Banach space β ◦ α (X, E), consists of all Baire functions with relatively compact ranges from a perfectly normal space X into a Banach space E. Moreover, we establish that if β ◦ α (X, E) is linear isometric with β ◦ α (Y,E), then some compactification of X and Y are homeomorphic. Keywords: Baire functions, Borel functions, Banach–Stone theorem, isomorphisms MSC 2000: Primary 26A21, 28A05, secondary 46B04, 54C50, 54H05 1. Introduction Let K be a compact Hausdorff space and E a Banach space. We designate by C (K, E) (resp. C (K )), the space of all E-valued (resp. real-valued) continuous functions on K , provided with sup-norm. It is well known that C (K ) ⊗ E is dense in C (K, E) ([8]). The Banach–Stone Theorem says that for two locally compact Hausdorff spaces X and Y , if T : C ◦ (X ) → C ◦ (Y ) is an isometric isomorphism, then there is a homeomorphism ϕ : Y → X and a continuous map u : Y →{λ ∈ R : |λ| =1} such that T (f )= u(f ◦ ϕ), for every f ∈ C ◦ (X ). Here C ◦ (X ) denotes the set of all real continuous functions on X which tends to zero at infinity ([2]). Some other versions of Banach–Stone Theorem have been proved by many authors (cf. [2, 4, 5, 11]). The Banach–Stone Theorem also has been generalized for the vector valued continuous functions (cf. [2]). The aim of this article is to replace the class of continuous functions in the above results with the class of Baire functions with relatively compact ranges and to establish similar results. For the remainder of this section, we introduce some definitions and basic facts. Throughout this paper, X and Y are two perfectly normal topological H. R. Shatery: Department of Mathematics, University of Isfahan, Iran; shatery@math.ui.ac.ir J. Zafarani: Department of Mathematics, University of Isfahan, Iran; jzaf@math.ui.ac.ir ISSN 0232-2064 / $ 2.50 c  Heldermann Verlag Berlin