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 $y \in Y$.

Proof 1

Since $Y < X$:

$X \setminus Y$ is non-empty.

Since $Y$ is closed:

$X \setminus Y$ is open.

Let $x \in X \setminus Y$.

Then there exists $\epsilon > 0$ such that:

$\map {B_\epsilon} x \subset X \setminus Y$

So, for all $y \in Y$, we must have:

$\norm {x - y} \ge \epsilon$

That is:

$\inf \set {\norm {x - y} \colon y \in Y} \ge \epsilon$

For brevity, let:

$d = \inf \set {\norm {x - y} \colon y \in Y}$

Since $\alpha^{-1} > 1$, there exists $z \in Y$ with:

$\norm {x - z} < d \alpha^{-1}$

otherwise the infimum would be at least $d \alpha^{-1} > d$, a contradiction.

Since $x \in X \setminus Y$ and $z \in Y$, we clearly have $x \ne z$.

So, we can set:

$ x_\alpha = \dfrac {x - z} {\norm {x - z} }$

Clearly:

$\norm {x_\alpha} = \dfrac {\norm {x - z} } {\norm {x - z} } = 1$

Now, for any $y \in Y$ we have:

\(\ds \norm {x_\alpha - y}\)

\(=\)

\(\ds \norm {\frac {x - z} {\norm {x - z} } - y}\)

\(\ds \)

\(=\)

\(\ds \frac {\norm {x - \paren {z + \norm {x - z} y} } } {\norm {x - z} }\)

We've already seen that:

$\norm {x - z} < d \alpha^{-1}$

Since $Y$ is closed under linear combination, we have:

$z + \norm {x - z} y \in Y$

and so:

$\norm {x - \paren {z + \norm {x - z} y} } \ge d$

We conclude that:

\(\ds \norm {x_\alpha - y}\)

\(=\)

\(\ds \frac {\norm {x - \paren {z + \norm {x - z} y} } } {\norm {x - z} }\)

\(\ds \)

\(>\)

\(\ds \frac d {d \alpha^{-1} }\)

\(\ds \)

\(=\)

\(\ds \alpha\)

$\blacksquare$

Proof 2

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:

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

$\blacksquare$

Source of Name

This entry was named for Frigyes Riesz.