mathematics

Mazur–Ulam theorem

Mazur–Ulam 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 Mazur–Ulam theorem rather than just read about it. In short: In mathematics, the Mazur–Ulam theorem states that if V {\displaystyle V} and W {\displaystyle W} are normed spaces over R and the mapping f : V → W {\displaystyle f\colon V\to W} is a surjective isometry, then f {\displaystyle f} is affine. It was proved by Stanisław Mazur and Stanisław Ulam in response to a question raised by Stefan Banach.

Key takeaways

Mazur–Ulam 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 Mazur–Ulam theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Mazur–Ulam theorem from memory before moving on to harder problems.

Reference excerpt

In mathematics, the Mazur–Ulam theorem states that if V {\displaystyle V} and W {\displaystyle W} are normed spaces over R and the mapping

f : V → W {\displaystyle f\colon V\to W}

is a surjective isometry, then f {\displaystyle f} is affine. It was proved by Stanisław Mazur and Stanisław Ulam in response to a question raised by Stefan Banach. For strictly convex spaces the result is true, and easy, even for isometries which are not necessarily surjective. In this case, for any u {\displaystyle u} and v {\displaystyle v} in V {\displaystyle V} , and for any t {\displaystyle t} in [ 0 , 1 ] {\displaystyle [0,1]} , write

r = ‖ u − v ‖ V = ‖ f ( u ) − f ( v ) ‖ W {\displaystyle r=\|u-v\|_{V}=\|f(u)-f(v)\|_{W}}

and denote the closed ball of radius R around v by B ¯ ( v , R ) {\displaystyle {\bar {B}}(v,R)} . Then t u + ( 1 − t ) v {\displaystyle tu+(1-t)v} is the unique element of B ¯ ( v , t r ) ∩ B ¯ ( u , ( 1 − t ) r ) {\displaystyle {\bar {B}}(v,tr)\cap {\bar {B}}(u,(1-t)r)} , so, since f {\displaystyle f} is injective, f ( t u + ( 1 − t ) v ) {\displaystyle f(tu+(1-t)v)} is the unique element of

f ( B ¯ ( v , t r ) ∩ B ¯ ( u , ( 1 − t ) r ) = f ( B ¯ ( v , t r ) ) ∩ f ( B ¯ ( u , ( 1 − t ) r ) = B ¯ ( f ( v ) , t r ) ∩ B ¯ ( f ( u ) , ( 1 − t ) r ) , {\displaystyle f{\bigl (}{\bar {B}}(v,tr)\cap {\bar {B}}(u,(1-t)r{\bigr )}=f{\bigl (}{\bar {B}}(v,tr){\bigr )}\cap f{\bigl (}{\bar {B}}(u,(1-t)r{\bigr )}={\bar {B}}{\bigl (}f(v),tr{\bigr )}\cap {\bar {B}}{\bigl (}f(u),(1-t)r{\bigr )},}

and therefore is equal to t f ( u ) + ( 1 − t ) f ( v ) {\displaystyle tf(u)+(1-t)f(v)} . Therefore f {\displaystyle f} is an affine map. This argument fails in the general case, because in a normed space which is not strictly convex two tangent balls may meet in some flat convex region of their boundary, not just a single point.

Worked examples

Example 1 — a first encounter with Mazur–Ulam theorem

Start with the simplest possible case. Write down what Mazur–Ulam 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 Mazur–Ulam 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 Mazur–Ulam 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 Mazur–Ulam theorem

In research: Mazur–Ulam 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 Mazur–Ulam 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: Mazur–Ulam theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Normed spaces, Theorems in functional analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Mazur–Ulam 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 Mazur–Ulam theorem in 20 minutes

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

Frequently asked questions

What is Mazur–Ulam theorem in simple terms?

In mathematics, the Mazur–Ulam theorem states that if V {\displaystyle V} and W {\displaystyle W} are normed spaces over R and the mapping f : V → W {\displaystyle f\colon V\to W} is a surjective isometry, then f {\displaystyle f} is affine. It was proved by Stanisław Mazur and Stanisław Ulam in re…

Why does Mazur–Ulam 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 Mazur–Ulam 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 Mazur–Ulam theorem.