mathematics

Mark Krein

Mark Krein 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 Mark Krein rather than just read about it. In short: Mark Grigorievich Krein (Russian: Марк Григо́рьевич Крейн; Ukrainian: Марко́ Григо́рович Крейн; 3 April 1907 – 17 October 1989) was a Soviet mathematician, one of the major figures of the Soviet school of functional analysis. He is known for works in operator theory (in close connection with concrete problems coming from mathematical physics), the problem of moments, classical analysis and representation theory.

Key takeaways

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

Reference excerpt

Mark Grigorievich Krein (Russian: Марк Григо́рьевич Крейн; Ukrainian: Марко́ Григо́рович Крейн; 3 April 1907 – 17 October 1989) was a Soviet mathematician, one of the major figures of the Soviet school of functional analysis. He is known for works in operator theory (in close connection with concrete problems coming from mathematical physics), the problem of moments, classical analysis and representation theory.

Early life and career He was born in Kiev, in the Russian Empire, leaving home at age 17 to go to Odessa. He had a difficult academic career, not completing his first degree and constantly being troubled by antisemitic discrimination. His supervisor was Nikolai Chebotaryov. He was awarded the Wolf Prize in Mathematics in 1982 (jointly with Hassler Whitney), but was not allowed to attend the ceremony. David Milman, Mark Naimark, Israel Gohberg, Vadym Adamyan, Mikhail Livsic and other known mathematicians were his students. He died in Odessa.

Legacy On 14 January 2008, the memorial plaque of Mark Krein was unveiled on the main administration building of Odesa University in Ukraine.

See also Tannaka–Krein duality Krein–Milman theorem and Krein–Rutman theorem in functional analysis Krein space Krein's condition for the indeterminacy of the problem of moments

External links O'Connor, John J.; Robertson, Edmund F., "Mark Krein", MacTutor History of Mathematics Archive, University of St Andrews Mark Krein at the Mathematics Genealogy Project INTERNATIONAL CONFERENCE Modern Analysis and Applications (MAA 2007). Dedicated to the centenary of Mark Krein Mark Krein at Find a Grave

Illustrations

Worked examples

Example 1 — a first encounter with Mark Krein

Start with the simplest possible case. Write down what Mark Krein 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 Mark Krein 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 Mark Krein 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 Mark Krein

In research: Mark Krein 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 Mark Krein 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: Mark Krein is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1907 births, 1989 deaths, 20th-century Ukrainian Jews, so understanding it makes those chapters shorter.
In everyday life: Look for Mark Krein 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 Mark Krein in 20 minutes

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

Frequently asked questions

What is Mark Krein in simple terms?

Why does Mark Krein 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 Mark Krein?

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 Mark Krein.