Three classical facts of the theory of normed spaces are considered: the boundedness principle, the Banach–Steinhaus theorem, and the uniform boundedness principle for a compact convex set. By means of Boolean valued analysis, exact analogs of the theorems are proven for the case of lattice-normed spaces.

Keywords:Banach–Steinhaus theorem, Banach–Kantorovich space, cyclically compact set, Boolean valued analysis.