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2 excluded silly counterexamples

Is the (algebraic) span a finite set of vectors in a Hausdorff topological vector space over a complete field always closed?

I suspect yes, but I can't come up with a proof, and it seems like locally convex might be needed to get this.

1

# Closedness of finite-dimensional subspaces

Is the (algebraic) span a finite set of vectors in a Hausdorff topological vector space always closed?

I suspect yes, but I can't come up with a proof, and it seems like locally convex might be needed to get this.