---
tag: 097C7KCAT5
kind: statement
role: lemma
name: Riesz's lemma
aliases: [riesz-lemma]
msc: [46B20]
uses: [0G5AV9Q4RR, 0J3578562S]
sources:
  - url: https://en.wikipedia.org/wiki/Riesz%27s_lemma
    locator: Statement
same_as:
  wikidata: Q1189968
  wikipedia: Riesz's lemma
---

## Statement

Let $X$ be a normed space over $\mathbb{R}$ or $\mathbb{C}$, and let $Y \neq X$ be a [[closed|0G5AV9Q4RR]] linear subspace of $X$.
Let $\alpha$ be a real number with $0 < \alpha < 1$.
Then there is $x \in X$ with $\|x\| = 1$ and $\|x - y\| \ge \alpha$ for all $y \in Y$.

## Remarks

Proof idea: take $x_0 \notin Y$ and let $r = \inf_{y \in Y} \|x_0 - y\| > 0$, which is positive because $Y$ is closed. Choose $y_0 \in Y$ with $\|x_0 - y_0\| \le r / \alpha$, and let $x = (x_0 - y_0) / \|x_0 - y_0\|$.
If $Y$ is finite-dimensional, one can take $\alpha = 1$.
The lemma is a substitute for orthogonal vectors in normed spaces without an inner product.
Its main application is that a [[compact closed unit ball forces finite dimension|0VZVV8VME5]].

## Explanations

### L1 human

For each closed proper subspace, there is a unit vector that is almost as far from the subspace as possible.
In an inner product space, a unit vector orthogonal to the subspace works with distance $1$.