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let $E$ be a banach space with infinite dimensional and $F$ a subspace of $E$. is it true that if $F^° \neq \emptyset $ then $E=F$.
my attempt: $F$ a subspace of $E$ then $F\subset E$ we have to prove that $E\subset F$ i think we should use the baire category theorem.

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No need for BCT. Some open ball $B(x,r) \subset F$. This implies $x\in F$ and $B(0,r) \subset F$. [$z \in B(0,r)$ implies $z+x \in B(x,r)$ so $z \in F$ . Also, $x\in F$ so $z=(z+x)-x \in F$]. Any $y$ can be written a $\frac 1 t (ty)$ and $ty \in B(0,r)\subset F$ if $t$ is suitably chosen. So $y \in F$.

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