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Theorem — Let X be a Banach space, C be a compact operator acting on X, and σ(C) be the spectrum of C. Every nonzero λ ∈ σ(C) is an eigenvalue of C. For all nonzero λ ∈ σ(C), there exist m such that Ker((λ − C) m) = Ker((λ − C) m+1), and this subspace is finite-dimensional. The eigenvalues can only accumulate at 0.
Imaginary depiction of the Library of Alexandria. The Pinakes (Ancient Greek: Πίνακες 'tables', plural of πίναξ pinax) is a lost bibliographic work composed by Callimachus (310/305–240 BCE) that is popularly considered to be the first library catalog in the West; its contents were based upon the holdings of the Library of Alexandria during Callimachus's tenure there during the ...
κ is weakly compact. for every λ<κ, natural number n ≥ 2, and function f: [κ] n → λ, there is a set of cardinality κ that is homogeneous for f. (Drake 1974, chapter 7 theorem 3.5) κ is inaccessible and has the tree property, that is, every tree of height κ has either a level of size κ or a branch of size κ.
Two volumes of a catalogue of the remaining works were published by Erich Hermann Mueller von Asow (1892–1964) in 1959. After von Asow's death, Franz Trenner (d. 1993) and Alfons Ott (d. 1976) published the third volume, based on von Asow's notes; this catalogue lists 323 titles, including Strauss's literary writings.
Compact group. The circle of center 0 and radius 1 in the complex plane is a compact Lie group with complex multiplication. In mathematics, a compact ( topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group).
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / ʃ ə ˈ l ɛ s k i / shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations.
In mathematics, a topological space X is sequentially compact if every sequence of points in X has a convergent subsequence converging to a point in . Every metric space is naturally a topological space, and for metric spaces, the notions of compactness and sequential compactness are equivalent (if one assumes countable choice ). However, there ...
Maximal compact subgroups of connected Lie groups are usually not unique, but they are unique up to conjugation, meaning that given two maximal compact subgroups K and L, there is an element g ∈ G such that gKg −1 = L. Hence a maximal compact subgroup is essentially unique, and people often speak of "the" maximal compact subgroup.