Distance and norm in a linear subspace

Question

Suppose $(X, \|\circ\|_X)$ is a normed space and $Y$ is a linear subspace of $X$ which is also normed space with respect to another norm $\|\circ\|_Y$. Is it possible that $\|x-y\|_Y < \infty$ for some $x \in X\backslash Y$ and $y \in Y$ ? Does this even make sens ? I'm new to this and I'm mixed up. Thanks.

Answer 1

It makes sense as long as $x-y$ is an element of $Y$, because the norm $\| \circ \|_Y$ is defined only on the elements of $Y$. However, if that is the case, then $x-y+y=x$ is an element of $Y$, which is a contradiction (because you chose your $x$ to be "outside" of $Y$).