I study Hilbert spaces and I have a question about dimension. If $G$ is a subspace of Hilbert space $H$. What is the relationship between $\text{dim}(G)$ and $\text{dim}(\text{cl}(G))$?
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Any finite dimensional subspace is closed. – shalop Feb 26 '17 at 12:43
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If $G$ is finite dimensional, then $G$ is already closed. If $G$ has countable dimension, then the dimension of $\overline{G}$ has strictly greater cardinality, because an infinite dimensional complete normed space must have uncountable dimension. If $G$ already has uncountable dimension, I am not sure what happens, let me think about it. – D_S Feb 26 '17 at 13:02