Let X₁ be a closed subspace and X₂ be a finite dimensional subspace of a normed space X. Then X₁ + X₂ is closed in X. Let X be a

Question

Let X₁ be a closed subspace and X₂ be a finite dimensional subspace of a normed space X. Then X₁ + X₂ is closed in X.
Let X be a normed space such that U(0,1) is totally bounded. Then X is finite dimensional.

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