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In the complementary subspace problem one asks for conditions on a locally convex vector space $E$ which guarantee that each closed subspace has a direct (topological) compliment in $E$. This question was studied for Banach spaces, and Lindenstrauss showed that a Banach space has this property if and only if it is a Hilbert space. Is there anything known for classes of LCVS beyond Banach spaces (e.g Frechet spaces, nuclear spaces) ?

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    $\begingroup$ For separable Frechet spaces, the questions is answered by Grothendieck (using Lindenstrauss' result). A separable Frechet space has "the complimented subspace property" if and only if it is either $\ell^{2}$, $\omega = (\mathbb{K})^{\mathbb{N}}$ or a product $\omega \ \times \ell^{2}$. $\endgroup$ – Sebastian Oct 23 '15 at 9:17

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