Linear and uniformly continuous surjections between <i>C_p</i>-spaces over metrizable spaces

For any Tychonoff space X let D(X) be either the set C(X) of all continuous functions on X or the set C*(X) of all bounded continuous functions on X. When D(X) is endowed with the pointwise convergence topology, we write Dp(X).Let T: Dp(X) -> Dp(Y) be a continuous linear surjection, where X is a metrizable space and Y is perfectly normal. We show that if X has some dimensional-like property P, then so does Y. For example, P could be one of the following properties: zero-dimensionality, countable-dimensionality or strong countable-dimensionality. This result remains true if T is a uniformly continuous and inversely bounded surjection.Also, we consider other properties P: of being a scattered space, or a strongly sigma-scattered space, or a Delta 1-space. Our results strengthen and extend several results from the various recently published papers.