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Post Closed as "too localized" by Yemon Choi, Alain Valette, Bill Johnson, Andreas Blass, quid
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2 | Fixed formulation according to comments.; added 11 characters in body | ||
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Suppose $(B,\|\cdot\|)$ is a Banach space, $V\subset B$ a dense subspace, and $V$ is equipped with a norm $\|\cdot\|_V$ such that $\|x\|_V = \|x\|$ for all $x\in V$. Is $(B,\|\cdot\|)$ the a completion of $(V,\|\cdot\|_V)$ with respect to the $\|\cdot\|_V$ topology? I.e., are can the norms identicalspaces be considered the same/identical up to some isometry? |
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Norms agreeing on dense subspaceSuppose $(B,\|\cdot\|)$ is a Banach space, $V\subset B$ a dense subspace, and $V$ is equipped with a norm $\|\cdot\|_V$ such that $\|x\|_V = \|x\|$ for all $x\in V$. Is $(B,\|\cdot\|)$ the completion of $(V,\|\cdot\|_V)$ with respect to the $\|\cdot\|_V$ topology? I.e., are the norms identical?
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