In the paper, it is proved that a topological space is a -space if and only if every image of the space under a Baire mapping to the Baire space is bounded. It is shown that there exists a compact -space such that its image under a Borel mapping to the Baire space is unbounded. The existence of such a space answers a question of L. Bukovsky and J. Haleš. Generalizations of results of N. N. Kholshchevnikova concerning the representation of functions on subsets of the number line by trigonometric series are obtained.