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17-585: Kadets may refer to: Mikhail Kadets , mathematician Members of the Constitutional Democratic Party , a political party in the Russian Empire See also [ edit ] Cadet (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Kadets . If an internal link led you here, you may wish to change

34-420: A class of random symmetric polytopes P ( ω ) {\displaystyle P(\omega )} in R n , {\displaystyle \mathbb {R} ^{n},} and the normed spaces X ( ω ) {\displaystyle X(\omega )} having P ( ω ) {\displaystyle P(\omega )} as unit ball (the vector space

51-595: A multiplicative constant independent from the dimension n {\displaystyle n} ). A major achievement in the direction of estimating the diameter of Q ( n ) {\displaystyle Q(n)} is due to E. Gluskin, who proved in 1981 that the (multiplicative) diameter of the Banach–Mazur compactum is bounded below by c n , {\displaystyle c\,n,} for some universal c > 0. {\displaystyle c>0.} Gluskin's method introduces

68-607: A projection of norm at most n . {\displaystyle {\sqrt {n}}.} Together with V. I. Gurarii and V. I. Matsaev, he found the exact order of magnitude of the Banach–Mazur distance between the n {\displaystyle n} -dimensional spaces ℓ p n {\displaystyle \ell _{p}^{n}} and ℓ q n . {\displaystyle \ell _{q}^{n}.} In harmonic analysis , Kadets proved (1964) what

85-468: Is R n {\displaystyle \mathbb {R} ^{n}} and the norm is the gauge of P ( ω ) {\displaystyle P(\omega )} ). The proof consists in showing that the required estimate is true with large probability for two independent copies of the normed space X ( ω ) . {\displaystyle X(\omega ).} Q ( 2 ) {\displaystyle Q(2)}

102-504: Is a Riesz basis in L 2 [ − π , π ] {\displaystyle L_{2}[-\pi ,\pi ]} Kadets was the founder of the Kharkov school of Banach spaces. Together with his son Vladimir Kadets, he authored two books about series in Banach spaces. Banach%E2%80%93Mazur distance In the mathematical study of functional analysis ,

119-624: Is now called the Kadets ; 1 / 4 {\displaystyle 1/4} theorem, which states that, if | λ n − n | ≤ C < 1 / 4 {\displaystyle |\lambda _{n}-n|\leq C<1/4} for all integers n , {\displaystyle n,} then the sequence ( exp ⁡ ( i λ n x ) ) n ∈ Z {\displaystyle (\exp(i\lambda _{n}x))_{n\in \mathbb {Z} }}

136-555: The Banach–Mazur distance is a way to define a distance on the set Q ( n ) {\displaystyle Q(n)} of n {\displaystyle n} -dimensional normed spaces . With this distance, the set of isometry classes of n {\displaystyle n} -dimensional normed spaces becomes a compact metric space , called the Banach–Mazur compactum . If X {\displaystyle X} and Y {\displaystyle Y} are two finite-dimensional normed spaces with

153-407: The dual space is separable. Together with Aleksander Pełczyński , he obtained important results on the topological structure of Lp spaces . Kadets also made several contributions to the theory of finite-dimensional normed spaces. Together with M. G. Snobar (1971), he showed that every n {\displaystyle n} -dimensional subspace of a Banach space is the image of

170-722: The State Prize of Ukraine in 2005. After reading the Ukrainian translation of Banach 's monograph Théorie des Opérations Linéaires , he became interested in the theory of Banach spaces. In 1966, Kadets solved in the affirmative the Banach – Fréchet problem, asking whether every two separable infinite-dimensional Banach spaces are homeomorphic . 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

187-485: The classical spaces, this upper bound for the diameter of Q ( n ) {\displaystyle Q(n)} is far from being approached. For example, the distance between ℓ n 1 {\displaystyle \ell _{n}^{1}} and ℓ n ∞ {\displaystyle \ell _{n}^{\infty }} is (only) of order n 1 / 2 {\displaystyle n^{1/2}} (up to

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204-422: The link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Kadets&oldid=873303487 " Categories : Disambiguation pages Disambiguation pages with surname-holder lists Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Mikhail Kadets Kadets

221-721: The maximal ellipsoid contained in a convex body gives the estimate: where ℓ n 2 {\displaystyle \ell _{n}^{2}} denotes R n {\displaystyle \mathbb {R} ^{n}} with the Euclidean norm (see the article on L p {\displaystyle L^{p}} spaces ). From this it follows that d ( X , Y ) ≤ n {\displaystyle d(X,Y)\leq n} for all X , Y ∈ Q ( n ) . {\displaystyle X,Y\in Q(n).} However, for

238-763: The maximum factor by which it "lengthens" vectors. The Banach–Mazur distance between X {\displaystyle X} and Y {\displaystyle Y} is defined by δ ( X , Y ) = log ⁡ ( inf { ‖ T ‖ ‖ T − 1 ‖ : T ∈ GL ⁡ ( X , Y ) } ) . {\displaystyle \delta (X,Y)=\log {\Bigl (}\inf \left\{\left\|T\right\|\left\|T^{-1}\right\|:T\in \operatorname {GL} (X,Y)\right\}{\Bigr )}.} We have δ ( X , Y ) = 0 {\displaystyle \delta (X,Y)=0} if and only if

255-402: The same dimension, let GL ⁡ ( X , Y ) {\displaystyle \operatorname {GL} (X,Y)} denote the collection of all linear isomorphisms T : X → Y . {\displaystyle T:X\to Y.} Denote by ‖ T ‖ {\displaystyle \|T\|} the operator norm of such a linear map —

272-1165: The spaces X {\displaystyle X} and Y {\displaystyle Y} are isometrically isomorphic. Equipped with the metric δ , the space of isometry classes of n {\displaystyle n} -dimensional normed spaces becomes a compact metric space , called the Banach–Mazur compactum. Many authors prefer to work with the multiplicative Banach–Mazur distance d ( X , Y ) := e δ ( X , Y ) = inf { ‖ T ‖ ‖ T − 1 ‖ : T ∈ GL ⁡ ( X , Y ) } , {\displaystyle d(X,Y):=\mathrm {e} ^{\delta (X,Y)}=\inf \left\{\left\|T\right\|\left\|T^{-1}\right\|:T\in \operatorname {GL} (X,Y)\right\},} for which d ( X , Z ) ≤ d ( X , Y ) d ( Y , Z ) {\displaystyle d(X,Z)\leq d(X,Y)\,d(Y,Z)} and d ( X , X ) = 1. {\displaystyle d(X,X)=1.} F. John's theorem on

289-502: Was born in Kiev. 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 in 1955 (under the supervision of Boris Levin ), and his doctoral dissertation in 1963. He was awarded

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