A note on the weak topology of spaces C_k(X) of continuous functions

It is well known that the property of being a bounded set in the class of topological vector spaces E is not a topological property, where a subset B⊂ E is called a bounded set if every neighbourhood of zero U in E absorbs B . The paper deals with the problem which topological properties of bounded sets for the space C_k(X) (of continuous real-valued functions on a Tychonoff space X with the compact-open topology) endowed with the weak topology of C_k(X) can be transferred to bounded sets of C_k(Y) endowed with the weak topology, assuming that the corresponding weak topologies of both C_k(X) and C_k(Y) are homeomorphic.