mathematics

Riesz's lemma

Riesz's lemma is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Riesz's lemma rather than just read about it. In short: In mathematics, Riesz's lemma (after Frigyes Riesz) is a lemma in functional analysis. It specifies (often easy to check) conditions that guarantee that a subspace in a normed vector space is dense.

Key takeaways

Riesz's lemma belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Riesz's lemma to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Riesz's lemma from memory before moving on to harder problems.

Reference excerpt

In mathematics, Riesz's lemma (after Frigyes Riesz) is a lemma in functional analysis. It specifies (often easy to check) conditions that guarantee that a subspace in a normed vector space is dense. The lemma may also be called the Riesz lemma or Riesz inequality. It can be seen as a substitute for orthogonality when the normed space is not an inner product space.

Statement

If X {\displaystyle X} is a reflexive Banach space then this conclusion is also true when α = 1. {\displaystyle \alpha =1.}

Metric reformulation As usual, let d ( x , y ) := ‖ x − y ‖ {\displaystyle d(x,y):=\|x-y\|} denote the canonical metric induced by the norm, call the set { x ∈ X : ‖ x ‖ = 1 } {\displaystyle \{x\in X:\|x\|=1\}} of all vectors that are a distance of 1 {\displaystyle 1} from the origin the unit sphere, and denote the distance from a point u {\displaystyle u} to the set Y ⊆ X {\displaystyle Y\subseteq X} by

d ( u , Y ) := inf y ∈ Y d ( u , y ) = inf y ∈ Y ‖ u − y ‖ . {\displaystyle d(u,Y)~:=~\inf _{y\in Y}d(u,y)~=~\inf _{y\in Y}\|u-y\|.} The inequality α ≤ d ( u , Y ) {\displaystyle \alpha \leq d(u,Y)} holds if and only if ‖ u − y ‖ ≥ α {\displaystyle \|u-y\|\geq \alpha } for all y ∈ Y , {\displaystyle y\in Y,} and it formally expresses the notion that the distance between u {\displaystyle u} and Y {\displaystyle Y} is at least α . {\displaystyle \alpha .} Because every vector subspace (such as Y {\displaystyle Y} ) contains the origin 0 , {\displaystyle 0,} substituting y := 0 {\displaystyle y:=0} in this infimum shows that d ( u , Y ) ≤ ‖ u ‖ {\displaystyle d(u,Y)\leq \|u\|} for every vector u ∈ X . {\displaystyle u\in X.} In particular, d ( u , Y ) ≤ 1 {\displaystyle d(u,Y)\leq 1} when ‖ u ‖ = 1 {\displaystyle \|u\|=1} is a unit vector. Using this new notation, the conclusion of Riesz's lemma may be restated more succinctly as: d ( u , Y ) ≥ α {\displaystyle d(u,Y)\geq \alpha } holds for some unit vector u ∈ X . {\displaystyle u\in X.}

Using this new terminology, Riesz's lemma may also be restated in plain English as:

Given any closed proper vector subspace of a normed space X , {\displaystyle X,} for any desired minimum distance α {\displaystyle \alpha } less than 1 , {\displaystyle 1,} there exists some vector in the unit sphere of X {\displaystyle X} that is at least this desired distance away from the subspace.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Riesz's lemma

Start with the simplest possible case. Write down what Riesz's lemma claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Riesz's lemma before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Riesz's lemma ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Riesz's lemma

In research: Riesz's lemma appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Riesz's lemma in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Riesz's lemma is common in secondary-school and first-year university syllabi. It links to neighbouring topics Functional analysis, Lemmas in mathematical analysis, Normed spaces, so understanding it makes those chapters shorter.
In everyday life: Look for Riesz's lemma outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Riesz's lemma in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Riesz's lemma means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Riesz's lemma out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Riesz's lemma in simple terms?

In mathematics, Riesz's lemma (after Frigyes Riesz) is a lemma in functional analysis. It specifies (often easy to check) conditions that guarantee that a subspace in a normed vector space is dense.

Why does Riesz's lemma matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Riesz's lemma?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Riesz's lemma.