Let $V$ be a locally convex vector space and let $U$ be a finite-dimensional subspace of $V$. The Hahn-Banach theorem guarantees that there exists a closed subspace $W$ of $V$ such that $$V=U\oplus W.$$
- Question 1: Is it true that direct sums and tensor products are distributive, i.e. do we have a direct sum decomposition $$V^{\otimes k}=U^{\otimes k}\oplus \left(U\otimes W^{\otimes (k-1)}\right)\oplus\cdots\oplus W^{\otimes k}?$$
- Question 2: Is it true, that this induces a decomposition of the complete projective tensor product, i.e. $$V^{\widehat\otimes k}=U^{\otimes k}\oplus \left(U\otimes W^{\widehat\otimes (k-1)}\right)\oplus\cdots\oplus W^{\widehat\otimes k}?$$
On the other hand $U^{\otimes k}$ is a finite-dimensional subspace of $V^{\widehat\otimes k}$ and hence there is a closed subspace $\widetilde W$ of $V^{\widehat\otimes k}$ such that $$V^{\widehat\otimes k}=U^{\otimes k}\oplus \widetilde W.$$
- Question 3: Can every such decomposition be induced by a decomposition of $V$ in the above sense?
- Question 4: Does anything change if we assume that $V$ is Banach?
