In this note subharmonic and plurisubharmonic functions on a complex space are studied intrinsically. For applications subharmonicity is characterized more effectually in terms of properties that need be verified only locally off a thin analytic subset; these include the submean-value inequalities, the spherical (respectively, solid) monotonicity, near as well as weak subharmonicity. Several results of Gunning [9, K and L] are extendable via regularity to complex spaces. In particular, plurisubharmonicity amounts (on a normal space) essentially to regularized weak plurisubharmonicity, and similarly for subharmonicity (on a general space). A generalized Hartogs’ lemma and constancy criteria for certain matrix-valued mappings are given.