How does Riesz's Theorem not contradict the existence of a unit sphere?

Question

Riesz's Theorem says that, for a non-dense subspace $X$ in a Banach space $Y$, given some $0<r<1$ there is some $y\in Y$ such that $|y|=1$ but $\inf_{x\in X}|x-y| > r$. However, it seems to me we could take $X$ to be the unit sphere in $Y = \mathbb{R}^\infty$, and now pick any $0<r<1$. It follows that there is some $|y|=1$ such that $\inf_{x\in X}|x-y|>r$ but also $y\in X$ so $|y-y|=0$. Can anyone explain my misunderstanding?