Riesz's Lemma

Theorem
Let $X$ be a normed vector space.

Let $Y$ be a proper closed linear subspace of $X$.

Let $\alpha \in \openint 0 1$.

Then there exists $x_\alpha \in X$ such that:


 * $\norm {x_\alpha} = 1$

with:


 * $\norm {x_\alpha - y} > \alpha$

for all $y \in Y$.