About dense subspaces in Banach spaces

Question

I have the following problem:

Let $E$ a Banach space and $X_1,X_2$ dense subspaces. Is $X_1\cap X_2$ dense in $X$? What is the answer if $X_1,X_2$ has codimension 1?

I don't know how to start. If anyone can give me a hint it will be appreciated.

Thanks in advance!

Answer 1

If $X_1$ and $X_2$ are dense, $X_1 \cap X_2$ need not be dense, in fact it might be just $\{0\}$. Consider any infinite-dimensional separable Banach space $E$. Start with a dense countable set $\{w_1, w_2, \ldots\}$. Inductively choose two sequences $x_j$ and $y_j$ such that $\|x_j - w_j\| < 1/j$ and $\|y_j - w_j\| < 1/j$ and $\{x_1, y_1, \ldots, x_j, y_j\}$ is linearly independent. Let $X_1$ be the linear span of the $x_j$ and $X_2$ the linear span of the $y_j$. Then $X_1$ and $X_2$ are dense in $E$ and $X_1 \cap X_2 = \{0\}$.