### On subspaces of locally convex spaces with unconditional Schauder bases

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Let (S, ∑, m) be any atomless finite measure space, and X any Banach space containing a copy of ${c}_{0}$. Then the Bochner space ${L}^{1}(m;X)$ is uncomplemented in ccabv(∑,m;X), the Banach space of all m-continuous vector measures that are of bounded variation and have a relatively compact range; and ccabv(∑,m;X) is uncomplemented in cabv(∑,m;X). It is conjectured that this should generalize to all Banach spaces X without the Radon-Nikodym property.

Fréchet spaces of strongly, weakly and weak*-continuous Fréchet space valued functions are considered. Complete solutions are given to the problems of their injectivity or embeddability as complemented subspaces in dual Fréchet spaces.

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