Let $X$ be a Banach space. $Y, X_1, X_2$ are 1-codimensional subspaces of X. $X_1, X_2$ are dense in X. Is $X_1 \cap X_2$ dense in X? Is $X1 \cap Y$ dense in $X$?
This is an exercise from Hahn-Banach theorem (chapter 3, Linear Analysis, Béla Bollobás). I tried to think $Y, X_1, X_2$ as kernels of linear functionals in order to apply Hahn-Banach theorem or hyperplane separation theorem but it doesn't work.
any help would be appreciated.