Free and projective generalized multinormed spaces

The paper investigates free and projective L-spaces, where L is a given normed space. These spaces form far-reaching generalization of known p-multinormed spaces; in particular, if L=Lp(X), the L-spaces can be considered as spaces, based on arbitrary σ-finite measure X (for “canonical” X=N with counting measure). We first describe “naturally appearing” functor, paving contractively complemented finite dimensional subspaces. This dimensionality essential; it permits us to L-space for this functor. As corollary, we obtain wide variety L-spaces. For “nice” (such space simple p-integrable functions space), full characterization L-spaces; particular case, recover description spaces.