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$Y_0\subset Y $ is a closed Hilbert subspace of $Y$ with finite codimension and a subspace $ Y_1 $ satisfies $ Y_0 \subset Y_1 \subset Y $. Is $ Y_1 $ also closed?

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Consider $\pi(Y_1) \subset Y/Y_{0}$, noting that a set is closed in the quotient topology precisely when its inverse is closed in the original space.

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