Is it possible that a countable dimension $\mathbb{C}$-algebra has a proper subalgebra of uncountable dimension? I think the answer is negative but I'm not sure.
2026-04-08 16:45:28.1775666728
Uncountable-dimension subalgebra of a countable dimension algebra
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No, it is not possible. This is unrelated to any $\mathbb{C}$-algebra structure and depends only on the basic linear algebra fact that if $W$ is a vector space and $V$ is a subspace of $W$, then $\dim(V)\leq\dim(W)$.