Free and projective generalized multinormed spaces

Abstract

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