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** The Banach – Alaoglu theorem about compactness of sets of functionals.

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Wikipedia

## Some Related Sentences

** and Banach

__**__

__Banach__algebra: an associative algebra A over the real or complex numbers which at the same time is also a

__Banach__space

**.**

** and –

__**__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

**.**

__**__Haliotis brazieri f

**.**hargravesi ( Cox, 1869 )

__–__synonym: Haliotis ethologus, the Mimic abalone, Haliotis hargravesi, the Hargraves ’ s abalone

__**__Haliotis diversicolor squamata Reeve, 1846

__–__synonym: Haliotis squamata

__–__the scaly Australian abalone

** and theorem

__**__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

**.**

__**__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

**.**

__**__

**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

**.**

__**__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

**.**

__**__Artin reciprocity law, a general

__theorem__in number theory that provided a partial solution to Hilbert's ninth problem

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