Right Type Departmental Bulletin Paper

Let $X $ be a space, $A $ a closed subspace of $X $ and $Z $ a locally convex linear topological space. Let $C(X, Z) $ be the linear space of all continuous mappings from $X $ to $Z $. A linear transformation $u:C(A, Z)arrow C(X, Z) $ is said to be a $Dugund_{\ddot{fl}} $ extender if $u $ satisfies the following conditions: For each $f\in C(A, Z) $, (a) $u(f) $ is an extension of $f $ , and (b) the range of $u(f) $ is contained in the closed convex hull of the range of $f $. The study of this area is initiated by Dugundji [2]. He proved that for every closed subspace $A $ of a metrizable space $X $ there exists a Dugundji extender $u $ : $C(A, \mathbb{R})arrow$ $C(X, \mathbb{R}) $. Michael ([8]) noticed that the Dugundji extender constructed by Dugundji is continous with respect to the pointwise convergence topology, the compact-open topology and the uniform convergence topology. We shall consider the Dugundji extention theorems on product spaces. Definition 1.1. Let $X $ be a space, $A $ a closed subspace of $X $ and $Z $ a locally convex linear topological space. Then we say that $A $ is $D(Z)$-embedded in $X $ if there is a Dugundji extender $u $ : $C(A, Z)arrow C(X, Z) $. Furthermore, we say that $A $ is $D$-embedded in $X $ if $A $ i