Assume that we have closed subspaces $Y_1$ and $Y_2$ of Banach spaces $X_1$ and $X_2$, respectively. If the product $Y_1\times Y_2$ is complemented in $X_1\times X_2$, does it follow that $Y_i$ is complemented in $X_i$ for $i=1, 2$.
1 Answer
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Yes. Let $P$ be a (continuous linear) projection from $X_1 \times X_2$ (equipped with the sum-norm, say) onto $Y_1 \times Y_2$. For $x_1\in X_1$ consider $P(x_1,0)$, which can be written in the form $(P_1x_1, P_2x_1) \in Y_1\times Y_2$. Then $P_1$ is a projection from $X_1$ onto $Y_1$.
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$\begingroup$ The answer (and maybe the question) implicitly assume that $Y_1\times Y_2$ is considered as subspace via the inclusion $i:(y_1,y_2)\mapsto (y_1,y_2)$ and uses a left inverse for $i$. If one allows other inclusions with closed range the answer is, in general, negative. $\endgroup$ Commented Jun 18 at 7:50