# Question in infinite- dimensional banach space.

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.

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 $$y \in F$$.