Free and projective generalized multinormed spaces

In this talk we investigate free and projective L-spaces, where L is a given normed space. The L-spaces are a far-reaching generalization of known $p$-multinormed spaces; in particular, in the case L:=$L_p(X)$ the L-spaces can be considered as $p$-multinormed spaces, based on arbitrary measure spaces $X$ (for “canonical” $p$-multinormed spaces, $X=\mathbb N$ with the counting measure).
First, we shall characterize free L-spaces, defined in terms on some naturally appearing functor. After this, using general-categorical applications of the freeness to the projectivity, we obtain a broad class of projective, which in “nice” cases (including the case of multinormed spaces) provides a full description of projective L-spaces.
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