Density question on intersection of two spaces

Question

Let $X$ and $Y$ be two Banach Spaces and $Z$ be a dense subset of $X$ and of $Y$, i.e., $\overline{(Z)}^{\|.\|_X} = X$ and $\overline{(Z)}^{\|.\|_Y} = Y$.

Then can we say that $Z$ is dense in $(X \cap Y, \|.\|_{X \cap Y})$, where $\|.\|_{X \cap Y}$ is any norm (to be specific either max norm or "$\|.\|_X + \|.\|_Y$" norm)?