mathematics

M. Riesz extension theorem

M. Riesz extension theorem 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 M. Riesz extension theorem rather than just read about it. In short: The M. Riesz extension theorem is a theorem in mathematics, proved by Marcel Riesz during his study of the problem of moments.

Key takeaways

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

Reference excerpt

The M. Riesz extension theorem is a theorem in mathematics, proved by Marcel Riesz during his study of the problem of moments.

Formulation Let E {\displaystyle E} be a real vector space, F ⊂ E {\displaystyle F\subset E} be a vector subspace, and K ⊂ E {\displaystyle K\subset E} be a convex cone. A linear functional ϕ : F → R {\displaystyle \phi :F\to \mathbb {R} } is called K {\displaystyle K} -positive, if it takes only non-negative values on the cone K {\displaystyle K} :

ϕ ( x ) ≥ 0 for x ∈ F ∩ K . {\displaystyle \phi (x)\geq 0\quad {\text{for}}\quad x\in F\cap K.}

A linear functional ψ : E → R {\displaystyle \psi :E\to \mathbb {R} } is called a K {\displaystyle K} -positive extension of ϕ {\displaystyle \phi } , if it is identical to ϕ {\displaystyle \phi } in the domain of ϕ {\displaystyle \phi } , and also returns a value of at least 0 for all points in the cone K {\displaystyle K} :

ψ | F = ϕ and ψ ( x ) ≥ 0 for x ∈ K . {\displaystyle \psi |_{F}=\phi \quad {\text{and}}\quad \psi (x)\geq 0\quad {\text{for}}\quad x\in K.}

In general, a K {\displaystyle K} -positive linear functional on F {\displaystyle F} cannot be extended to a K {\displaystyle K} -positive linear functional on E {\displaystyle E} . Already in two dimensions one obtains a counterexample. Let E = R 2 , K = { ( x , y ) : y > 0 } ∪ { ( x , 0 ) : x > 0 } , {\displaystyle E=\mathbb {R} ^{2},\ K=\{(x,y):y>0\}\cup \{(x,0):x>0\},} and F {\displaystyle F} be the x {\displaystyle x} -axis. The positive functional ϕ ( x , 0 ) = x {\displaystyle \phi (x,0)=x} can not be extended to a positive functional on E {\displaystyle E} . However, the extension exists under the additional assumption that E ⊂ K + F , {\displaystyle E\subset K+F,} namely for every y ∈ E , {\displaystyle y\in E,} there exists an x ∈ F {\displaystyle x\in F} such that y − x ∈ K . {\displaystyle y-x\in K.}

Proof The proof is similar to the proof of the Hahn–Banach theorem (see also below). By transfinite induction or Zorn's lemma it is sufficient to consider the case dim E / F = 1 {\displaystyle E/F=1} . Choose any y ∈ E ∖ F {\displaystyle y\in E\setminus F} . Set

a = sup { ϕ ( x ) ∣ x ∈ F , y − x ∈ K } , b = inf { ϕ ( x ) ∣ x ∈ F , x − y ∈ K } . {\displaystyle a=\sup\{\,\phi (x)\mid x\in F,\ y-x\in K\,\},\ b=\inf\{\,\phi (x)\mid x\in F,x-y\in K\,\}.}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with M. Riesz extension theorem

Start with the simplest possible case. Write down what M. Riesz extension theorem 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 M. Riesz extension theorem 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 M. Riesz extension theorem 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 M. Riesz extension theorem

In research: M. Riesz extension theorem 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 M. Riesz extension theorem 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: M. Riesz extension theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Theorems in convex geometry, Theorems in functional analysis, so understanding it makes those chapters shorter.
In everyday life: Look for M. Riesz extension theorem 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 M. Riesz extension theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what M. Riesz extension theorem 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 M. Riesz extension theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is M. Riesz extension theorem in simple terms?

The M. Riesz extension theorem is a theorem in mathematics, proved by Marcel Riesz during his study of the problem of moments.

Why does M. Riesz extension theorem 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 M. Riesz extension theorem?

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 M. Riesz extension theorem.