Normed and Banach Spaces - Applications

The Riesz Lemma provides a fundamental characterization of finite-dimensional subspaces in normed spaces, revealing a crucial difference between finite and infinite dimensions.

TheoremRiesz Lemma

Let $X$ be a normed space and $Y \subsetneq X$ a proper closed subspace. Then for every $\varepsilon \in (0,1)$ , there exists $x_\varepsilon \in X$ with $\|x_\varepsilon\| = 1$ such that $\|x_\varepsilon - y\| \geq 1 - \varepsilon$ for all $y \in Y$ .

This lemma shows that in any normed space, we can find unit vectors that are "almost orthogonal" to any proper closed subspace. The geometric intuition is that closed subspaces have "room to spare" for vectors far from them.

Proof

Since $Y$ is a proper subspace, there exists $x_0 \in X \setminus Y$ . Because $Y$ is closed, the distance $d = \inf_{y \in Y} \|x_0 - y\|$ is positive. Choose $y_0 \in Y$ such that $\|x_0 - y_0\| < \frac{d}{1-\varepsilon}$ .

Define $x_\varepsilon = \frac{x_0 - y_0}{\|x_0 - y_0\|}$

Then $\|x_\varepsilon\| = 1$ . For any $y \in Y$ , we have $y_0 + \|x_0 - y_0\| y \in Y$ , so $\|x_\varepsilon - y\| = \frac{1}{\|x_0 - y_0\|} \|x_0 - y_0 - \|x_0 - y_0\| y\|$ $= \frac{1}{\|x_0 - y_0\|} \|x_0 - (y_0 + \|x_0 - y_0\| y)\| \geq \frac{d}{\|x_0 - y_0\|} > 1 - \varepsilon$

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ExampleApplication to Compactness

A key application of the Riesz Lemma is characterizing when the closed unit ball is compact:

Theorem: In a normed space $X$ , the closed unit ball $\overline{B} = \{x \in X : \|x\| \leq 1\}$ is compact if and only if $\dim X < \infty$ .

Proof Sketch: If $\dim X = \infty$ , we can construct an infinite sequence in $\overline{B}$ with no convergent subsequence using the Riesz Lemma repeatedly. Start with any $x_1$ with $\|x_1\| = 1$ , let $Y_1 = \text{span}\{x_1\}$ , and apply Riesz Lemma with $\varepsilon = 1/2$ to find $x_2$ with $\|x_2\| = 1$ and $\|x_2 - x_1\| \geq 1/2$ . Continue inductively.

Remark

The Riesz Lemma is "almost" an orthogonality result. In Hilbert spaces, we can achieve exact orthogonality ( $\varepsilon = 0$ ) using the projection theorem. The lemma shows that even without an inner product structure, closed subspaces of normed spaces behave somewhat like orthogonal complements.

This theorem is fundamental to understanding the geometry of infinite-dimensional spaces and appears in the proofs of many important results, including the characterization of compact operators and the spectral theory of bounded linear operators.