Mikhail Iosiphovich Kadets (, , sometimes transliterated as Kadec, 30 November 1923 – 7 March 2011) was a Soviet-born Jewish mathematician working in

analysis Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...

and the theory of

Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...

Life and work

Kadets was born in

Kyiv Kyiv, also Kiev, is the capital and most populous List of cities in Ukraine, city of Ukraine. Located in the north-central part of the country, it straddles both sides of the Dnieper, Dnieper River. As of 1 January 2022, its population was 2, ...

. In 1943, he was drafted into the army. After demobilisation in 1946, he studied at Kharkov University, graduating in 1950. After several years in Makeevka he returned to Kharkov in 1957, where he spent the remainder of his life working at various institutes. He defended his

PhD A Doctor of Philosophy (PhD, DPhil; or ) is a terminal degree that usually denotes the highest level of academic achievement in a given discipline and is awarded following a course of graduate study and original research. The name of the deg ...

in 1955 (under the supervision of Boris Levin), and his doctoral dissertation in 1963. He was awarded the State Prize of Ukraine in 2005. After reading the Ukrainian translation of

Banach Banach (pronounced in German, in Slavic Languages, and or in English) is a Jewish surname of Ashkenazi origin believed to stem from the translation of the phrase "Son of man (Judaism), son of man", combining the Hebrew language, Hebrew word ' ...

's monograph '' Théorie des Opérations Linéaires'', he became interested in the theory of Banach spaces. In 1966, Kadets solved in the affirmative the

– Fréchet problem, asking whether every two separable infinite-dimensional Banach spaces are

homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...

. He developed the method of equivalent norms, which has found numerous applications. For example, he showed that every separable Banach space admits an equivalent Fréchet differentiable norm if and only if the

dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V,'' together with the vector space structure of pointwise addition and scalar multiplication by cons ...

is separable. Together with Aleksander Pełczyński, he obtained important results on the topological structure of

Lp space In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourba ...

s. Kadets also made several contributions to the theory of finite-dimensional normed spaces. Together with M. G. Snobar (1971), he showed that every

n

-dimensional subspace of a Banach space is the image of a projection of norm at most

\sqrt.

Together with V. I. Gurarii and V. I. Matsaev, he found the exact order of magnitude of the Banach–Mazur distance between the

n

-dimensional spaces

\ell^n_p

and

\ell^n_q.

harmonic analysis Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...

, Kadets proved (1964) what is now called the Kadets

1/4

theorem, which states that, if

, \lambda_n - n,  \leq C < 1/4

for all integers

n,

then the sequence

(\exp(i \lambda_n x))_

is a Riesz basis in

L_2 \pi, \pi /math>

Kadets was the founder of the

Kharkov Kharkiv, also known as Kharkov, is the second-largest List of cities in Ukraine, city in Ukraine.

school of Banach spaces. Together with his son Vladimir Kadets, he authored two books about series in Banach spaces.

Notes

External links

Kadets memorial website
* {{DEFAULTSORT:Kadets, Mikhail Functional analysts Mathematical analysts National University of Kharkiv alumni 1923 births 2011 deaths Scientists from Kyiv Soviet mathematicians Ukrainian mathematicians Soviet Jews Ukrainian Jews Laureates of the State Prize of Ukraine in Science and Technology