On the Quasinormal Convergence of Functions

In the paper, it is proved that a topological space $X$ is a $QN$-space if and only if every image of the space $X$ under a Baire mapping to the Baire space $\mathbb{N}^{\mathbb{N}}$ is bounded. It is shown that there exists a compact $QN$-space such that its image under a Borel mapping to the Baire space $\mathbb{N}^{\mathbb{N}}$ is unbounded. The existence of such a space answers a question of L. Bukovský 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.