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Page "Axiom of choice" ¶ 97
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** and Banach
** The Banach Tarski paradox.
** The Hahn Banach theorem in functional analysis, allowing the extension of linear functionals
** Banach algebra: an associative algebra A over the real or complex numbers which at the same time is also a Banach space.
** c < sub > 0 </ sub >, the Banach space of real sequences that converge to zero

** and
** Atlas III was a US launch vehicle ( 2000 2005 )
** Atlas V ( 2002 Present )
** Atlas Major, a Standard Motor Company van 1962 1963
** The Nielsen Schreier theorem, that every subgroup of a free group is free.
** Every Tychonoff space has a Stone Čech compactification.
** This is a distinguished work which stands out from, and above, many of the books and articles which have ben written in this century on Avicenna ( Ibn Sīnā ) ( A. D. 980 1037 ).
** The numbers and are not algebraic numbers ( see the Lindemann Weierstrass theorem ); hence they are transcendental.
** Laozi ( 5th 4th century BC )
** Ge Hong ( 283 AD 343 AD )
** Xun Zi ( c. 312 BC 230 BC )
** Gongsun Long ( c. 325 BC c. 250 BC )
** Zou Yan ( 305 BC 240 BC )
** Mani ( c. 216 AD 276 AD )
** Haliotis brazieri f. hargravesi ( Cox, 1869 ) synonym: Haliotis ethologus, the Mimic abalone, Haliotis hargravesi, the Hargraves ’ s abalone
** Haliotis dalli roberti McLean, 1970 synonym: Haliotis roberti
** Haliotis diversicolor squamata Reeve, 1846 synonym: Haliotis squamata the scaly Australian abalone
** Haliotis diversicolor supertexta the Taiwan abalone or jiukong
** Haliotis kamtschatkana assimilis Dall, 1878 synonym: Haliotis assimilis, the threaded abalone
** Haliotis ovina f. patamakanthini Dekker, Regter, & Gras, 2001 synonym: Haliotis patamakanthini
** Haliotis rubra conicopora Péron, 1816 synonym: Haliotis conicopora the conical pore abalone

** and theorem
** Well-ordering theorem: Every set can be well-ordered.
** Tarski's theorem: For every infinite set A, there is a bijective map between the sets A and A × A.
** König's theorem: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger cardinals.
** Tychonoff's theorem stating that every product of compact topological spaces is compact.
** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.
** The Vitali theorem on the existence of non-measurable sets which states that there is a subset of the real numbers that is not Lebesgue measurable.
** Stone's representation theorem for Boolean algebras needs the Boolean prime ideal theorem.
** The theorem that every Hilbert space has an orthonormal basis.
** The Baire category theorem about complete metric spaces, and its consequences, such as the open mapping theorem and the closed graph theorem.
** Gödel's completeness theorem for first-order logic: every consistent set of first-order sentences has a completion.
** Hilbert's basis theorem
** Bayes ' theorem
** More generally, Rademacher's theorem extends the differentiability result to Lipschitz mappings between Euclidean spaces: a Lipschitz map ƒ: U → R < sup > m </ sup >, where U is an open set in R < sup > n </ sup >, is almost everywhere differentiable.
** Lyapunov's central limit theorem
** Superposition theorem, in electronics
** " Kelvin's vorticity theorem for incompressible or barotropic flow ".
** Artin reciprocity law, a general theorem in number theory that provided a partial solution to Hilbert's ninth problem
** Various proofs of the four colour theorem.

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