In this note we prove the equivalence of the pointwise constancy and the global constancy of the holomorphic sectional curvature of a $K$-space. A criterion for the constancy of the holomorphic sectional curvature of a $K$-space is found. It is proved that every proper $K$-space of constant holomorphic sectional curvature is a six-dimensional orientable Riemannian manifold of constant positive curvature, which is isometric with the six-dimensional sphere in the case of completeness and connectedness.