The algebraic dimension of linear metric spaces and Baire properties of their hyperspaces

Answering a question of Halbeisen we prove (by two different methods) that the algebraic dimension of each infinite-dimensional complete linear metric space X equals the size of X. A topological method gives a bit more: the algebraic dimension of a linear metric space X equals jXj provided the hyperspace K(X) of compact subsets of X is a Baire space. Studying the interplay between Baire properties of a linear metric space X and its hyperspace, we construct a hereditarily Baire linear metric space X with meager hyperspace K(X). Also under (d = c) we construct a metrizable separable noncomplete linear metric space with hereditarily Baire hyperspace. We do not know if such a space can be constructed in ZFC.