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Let $Y$ be a vector space. Let $Z$ be a linear subspace of $Y$. Suppose that $Y/Z$ is finite dimensional. Does it follows that $Y/W$ is finite dimensional if $W$ is isomorphic to $Z$ as a vector space?

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up vote 8 down vote accepted

No. Take a vector space $Y$ with basis $e_1,e_2,e_3,\ldots$ and let $W=\textrm{span}(e_2,e_4,e_6,\ldots)$ and $Z=Y$. Clearly, $W$ is isomorphic to $Z$ as a vector space, but $Y/Z$ is trivial while $Y/W$ is infinite-dimensional.

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