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** The theorem that every Hilbert space has an orthonormal basis.

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## Some Related Sentences

** 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

**.**

__**__

**The**numbers and are not algebraic numbers ( see the Lindemann – Weierstrass

__theorem__); hence they are transcendental

**.**

__**__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

** and every

__**__Zorn's lemma: Every non-empty partially ordered set in which

__every__chain ( i

**.**e

**.**totally ordered subset )

**has**

**an**upper bound contains at least one maximal element

**.**

__**__Hausdorff maximal principle: In any partially ordered set,

__every__totally ordered subset is contained in a maximal totally ordered subset

**.**

__**__

**The**microcode can employ both pipelines to enable auto-repeating instructions such as rep movsw perform one iteration

__every__clock cycle, while the 80486 needed three clocks per iteration ( and the earliest x86-chips significantly more than the 486 ).

__**__United Kingdom Census held, the first to record names and approximate ages of

__every__household member and to be administered nationally

**.**

__**__Circuit analysis, the process of finding the voltages across, and the currents through,

__every__component in

**an**electrical circuit

__**__Unique factorization domain,

**an**integral domain in which

__every__non-zero element can be written as a product of irreducible elements in essentially a unique way

__**__Atomic domain,

**an**integral domain in which

__every__non-zero non-unit is a finite product of irreducible elements

__**__Dedekind domain,

**an**integral domain in which

__every__nonzero proper ideal factors into a product of prime ideals

__**__GCD domain,

**an**integral domain in which

__every__two non-zero elements have a greatest common divisor

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