In constructive set theory, however, Diaconescu's theorem shows that the axiom of choice implies the law of the excluded middle ( unlike in Martin-Löf type theory, where it does not ).

The first major application was the relative version of Serre's theorem showing that the cohomology of a coherent sheaf on a complete variety is finite dimensional ; Grothendieck's theorem shows that the higher direct images of coherent sheaves under a proper map are coherent ; this reduces to Serre's theorem over a one-point space.

A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the set of real numbers and the set of natural numbers do not have the same cardinal number.

This incompleteness result is similar to Gödel's incompleteness theorem in that it shows that no consistent formal theory for arithmetic can be complete.

As with the finiteness theorem, he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers.

Animation illustrating Pythagorean theorem | Pythagoras ' rule for a right-angle triangle, which shows the algebraic relationship between the triangle's hypotenuse, and the other two sides.

The Gauss-Markov theorem shows that the OLS estimator is the best ( minimum variance ), unbiased estimator assuming the model is linear, the expected value of the error term is zero, errors are homoskedastic and not autocorrelated, and there is no perfect multicollinearity.

The Bohr – van Leeuwen theorem shows that magnetism cannot occur in purely classical solids.

Gödel's incompleteness theorem, another celebrated result, shows that there are inherent limitations in what can be achieved with formal proofs in mathematics.

Therefore, just as Bayes ' theorem shows, the result of each trial comes down to the base probability of the fair coin:.

Shannon's entropy represents an absolute limit on the best possible lossless compression of any communication, under certain constraints: treating messages to be encoded as a sequence of independent and identically distributed random variables, Shannon's source coding theorem shows that, in the limit, the average length of the shortest possible representation to encode the messages in a given alphabet is their entropy divided by the logarithm of the number of symbols in the target alphabet.

The fundamental theorem of arithmetic guarantees that there is only one possible string that will be accepted ( providing the factors are required to be listed in order ), which shows that the problem is in both UP and co-UP.

However, even though it cannot be determined whether a particular file is incompressible, a simple theorem about incompressible strings shows that over 99 % of files of any given length cannot be compressed by more than one byte ( including the size of the decompressor ).

Gödel's second incompleteness theorem ( 1931 ) shows that no formal system extending basic arithmetic can be used to prove its own consistency.

While crystals, according to the classical crystallographic restriction theorem, can possess only two, three, four, and six-fold rotational symmetries, the Bragg diffraction pattern of quasicrystals shows sharp peaks with other symmetry orders, for instance five-fold.

For example, Rice's theorem shows that each of the following sets of computable functions is undecidable:

Gödel's first incompleteness theorem shows that for languages sufficient for doing a certain amount of arithmetic, there can be no effective deductive system that is complete with respect to the intended interpretation of the symbolism of that language.

The postulate is justified in part, for classical systems, by Liouville's theorem ( Hamiltonian ), which shows that if the distribution of system points through accessible phase space is uniform at some time, it remains so at later times.

For example, the Fitting lemma shows that the endomorphism ring of a finite length indecomposable module is a local ring, so that the strong Krull-Schmidt theorem holds and the category of finite length modules is a Krull-Schmidt category.

The Sonnenschein – Mantel – Debreu theorem shows that the standard model cannot be rigorously derived in general from general equilibrium theory.

Another theorem shows that there are problems solvable by Turing-complete languages that cannot be solved by any language with only finite looping abilities ( i. e., any language that guarantees every program will eventually finish to a halt ).

Gödel's incompleteness theorem shows that no consistent, recursively enumerable theory ( that is, one whose theorems form a recursively enumerable set ) in which the concept of natural numbers can be expressed, can include all true statements about them.

The theorem also shows that any group of prime order is cyclic and simple.

In mathematics terminology, the vector space of bras is the dual space to the vector space of kets, and corresponding bras and kets are related by the Riesz representation theorem.

* Duality: Every statement, theorem, or definition in category theory has a dual which is essentially obtained by " reversing all the arrows ".

A sufficient condition for recovering s ( t ) ( and therefore S ( ƒ )) from just these samples is that the non-zero portion of s ( t ) be confined to a known interval of duration P, which is the frequency domain dual of the Nyquist – Shannon sampling theorem.

This theorem establishes an important connection between a Hilbert space and its ( continuous ) dual space: if the underlying field is the real numbers, the two are isometrically isomorphic ; if the field is the complex numbers, the two are isometrically anti-isomorphic.

The following theorem, also referred to as the Riesz-Markov theorem, gives a concrete realisation of the dual space of C < sub > 0 </ sub >( X ), the set of continuous functions on X which vanish at infinity.

There is a dual existence theorem for colimits in terms of coequalizers and coproducts.

This allows us to use the Riesz representation theorem and find that the dual space of

These include theorems about compactness of certain spaces such as the Banach – Alaoglu theorem on the compactness of the unit ball of the dual space of a normed vector space, and the Arzelà – Ascoli theorem characterizing the sequences of functions in which every subsequence has a uniformly convergent subsequence.

The dual theorem holds for the greatest fixpoint.

The concept of dual cell structure, which Henri Poincaré used in his proof of his Poincaré duality theorem, contained the germ of the idea of cohomology, but this was not seen until later.

Every finite-dimensional normed space is reflexive, simply because in this case, the space, its dual and bidual all have the same linear dimension, hence the linear injection J from the definition is bijective, by the rank-nullity theorem.

The general form of the Plancherel theorem tries to describe the regular representation of G on L < sup > 2 </ sup >( G ) by means of a measure on the unitary dual.

For G compact, this is done by the Peter-Weyl theorem ; in that case the unitary dual is a discrete space, and the measure attaches an atom to each point of mass equal to its degree.

Hence, DLat is dual to CohSp-one obtains Stone's representation theorem for distributive lattices.

One can define the Lie algebra of an algebraic group purely algebraically ( it consists of the dual number points based at the identity element ); and this theorem shows that we get a matrix Lie algebra.

In functional analysis and related branches of mathematics, the Banach – Alaoglu theorem ( also known as Alaoglu's theorem ) states that the closed unit ball of the dual space of a normed vector space is compact in the weak * topology.

A special case of the Banach – Alaoglu theorem is the sequential version of the theorem, which asserts that the closed unit ball of the dual space of a separable normed vector space is sequentially compact in the weak * topology.

The importance of this simple definition stems from the fact that each and every definition and theorem of order theory can readily be transferred to the dual order.

This means that for every theorem of classical logic there is an equivalent dual theorem.

When the measure space is furthermore sigma-finite then L < sup >∞</ sup >( μ ) is in turn dual to L < sup > 1 </ sup >( μ ), which by the Radon – Nikodym theorem is identified with the set of all countably additive μ-a. c. measures.