Riesz's Lemma/Proof 2

Proof
Consider the normed quotient vector space $X / Y$ with quotient mapping $\pi$.

From Operator Norm of Quotient Mapping in Quotient Normed Vector Space is 1, we have:


 * $\norm \pi_{\map B {X, X/Y} } = 1$

Since $\alpha \in \openint 0 1$, there exists $x_\alpha \in X$ with $\norm {x_\alpha} = 1$ and:


 * $\norm {\map \pi {x_\alpha} }_{X/Y} > \alpha$

by the definition of the norm on the space of bounded linear transformations.

That is, by the definition of the quotient norm:


 * $\ds \inf_{z \mathop \in Y} \norm {x_\alpha - z} > \alpha$

So there exists $y \in Y$ such that:


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