Every proper subspace of a Banach space is either closed or dense

Question

Let $E$ be Banach, $F$ a proper subspace, take $y$ outside $F$, define $f(x+ty)=t$ for $x$ in $F$ and $t$ real, it is a linear functional and the kernel is $F$. Then $F$ not dense iff $F$ closed in $E$ iff $f$ is bounded. Is it correct to say I can't have a proper not dense not closed subspace? Thank you.

Answer 1

Ohh okay , i thought that you extended your functional in the whole space E by using Hahn-Banach. If you extend the functional in the whole space E then you might have F⊊ kernel. If you dont extend your functional is defined only in span(F∪{y}) which might not be the whole E so you cant use the preposition which says F not dense in E iff F closed in E iff f is bounded.This prepositions holds for the space span(F∪{y}) . I hope this is better !