The theorem which we prove is more general than what we have described, since it works with the primary decomposition of the minimal polynomial, whether or not the primes which enter are all of first degree.

In the primary decomposition theorem, it is not necessary that the vector space V be finite dimensional, nor is it necessary for parts ( A ) and ( B ) that P be the minimal polynomial for T.

According to Cauchy's functional equation theorem, the logarithm is the only continuous transformation that transforms real multiplication to addition.

One motivation for this use is that a number of generally accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs.

However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.

The debate is interesting enough, however, that it is considered of note when a theorem in ZFC ( ZF plus AC ) is logically equivalent ( with just the ZF axioms ) to the axiom of choice, and mathematicians look for results that require the axiom of choice to be false, though this type of deduction is less common than the type which requires the axiom of choice to be true.

In Martin-Löf type theory and higher-order Heyting arithmetic, the appropriate statement of the axiom of choice is ( depending on approach ) included as an axiom or provable as a theorem.

Assuming ZF is consistent, Kurt Gödel showed that the negation of the axiom of choice is not a theorem of ZF by constructing an inner model ( the constructible universe ) which satisfies ZFC and thus showing that ZFC is consistent.

Assuming ZF is consistent, Paul Cohen employed the technique of forcing, developed for this purpose, to show that the axiom of choice itself is not a theorem of ZF by constructing a much more complex model which satisfies ZF ¬ C ( ZF with the negation of AC added as axiom ) and thus showing that ZF ¬ C is consistent.

It is possible, however, that there is a shorter proof of a theorem from ZFC than from ZF.

** 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.

A more general binomial theorem and the so-called " Pascal's triangle " were known in the 10th-century A. D. to Indian mathematician Halayudha and Persian mathematician Al-Karaji, in the 11th century to Persian poet and mathematician Omar Khayyam, and in the 13th century to Chinese mathematician Yang Hui, who all derived similar results .< ref > Al-Karaji also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using mathematical induction.

In the 1880s, it became clear that results similar to the Bolzano – Weierstrass theorem could be formulated for spaces of functions rather than just numbers or geometrical points.

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

The Stolz-Cesàro theorem is a similar result involving limits of sequences, but it uses finite difference operators rather than derivatives.

This is similar ( but more categorical ) to concepts in group theory or module theory, where a given decomposition of an object into a direct sum is " not natural ", or rather " not unique ", as automorphisms exist that do not preserve the direct sum decomposition – see Structure theorem for finitely generated modules over a principal ideal domain # Uniqueness for example.

In his work on the divine proportion, H. E. Huntley relates the feeling of reading and understanding someone else's proof of a theorem of mathematics to that of a viewer of a masterpiece of art — the reader of a proof has a similar sense of exhilaration at understanding as the original author of the proof, much as, he argues, the viewer of a masterpiece has a sense of exhilaration similar to the original painter or sculptor.

For example, Euclid provides an elaborate proof of the Pythagorean theorem, by using addition of areas instead of the much simpler proof from similar triangles, which relies on ratios of line segments.

A similar derivation applies for conditioning on multiple events, using the appropriate extension of Bayes ' theorem

The application of De Morgan's theorem to a conjunction is very similar to its application to a disjunction both in form and rationale.

* U is diagonalizable ; that is, U is unitarily similar to a diagonal matrix, as a consequence of the spectral theorem.

This notation is very similar to usual base-n positional notation, but the usual notation does not suffice for the purposes of Goodstein's theorem.

The theorem is very similar to Menelaus ' theorem in that their equations differ only in sign.

This restriction is similar to the restriction to continuous operators in the Kleene fixed-point theorem of order theory.

A theorem similar to van der Waerden's theorem is Schur's theorem: for any given c there is a number N such that if the numbers 1, 2, ..., N are coloured with c different colours, then there must be a pair of integers x, y

The spectral theorem says that every normal matrix is unitarily similar to a diagonal matrix ( if AA < sup >*</ sup > = A < sup >*</ sup > A then there exists a unitary matrix U such that UAU < sup >*</ sup > is diagonal ).

Several theorems in number theory similar to Fermat's Last Theorem follow from the modularity theorem.

A similar theorem is not known for time-bounded probabilistic complexity classes, unless the class also has advice.

* The relationship between the general and extended form of Brahmagupta's formula is similar to how the law of cosines extends the Pythagorean theorem.

Other results include the Markov – Kakutani fixed-point theorem ( 1936-1938 ) and the Ryll-Nardzewski fixed-point theorem ( 1967 ) for continuous affine self-mappings of compact convex sets, as well as the Earle – Hamilton fixed-point theorem ( 1968 ) for holomorphic self-mappings of open domains.

* Kakutani fixed point theorem

For locally compact abelian groups, this property is satisfied as a result of the Markov – Kakutani fixed-point theorem.

* Kakutani fixed-point theorem

The Kakutani fixed-point theorem is a generalization of Brouwer's fixed-point theorem, holding for generalized correspondences instead of functions.

* Kakutani fixed point theorem

In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions.

The Kakutani fixed point theorem is a generalization of Brouwer fixed point theorem.

The theorem was developed by Shizuo Kakutani in 1941 and was famously used by John Nash in his description of Nash equilibria.

The latter theorem has been generalized by Yamabe and Yujobo, and Cairns to show that in Af there are families of such cubes.

** 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.

Note that " completeness " has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement such that neither nor can be proved from the given set of axioms.

The theorem says that, among algorithms that decode strings from their descriptions ( codes ), there exists an optimal one.

* Metamath-a language for developing strictly formalized mathematical definitions and proofs accompanied by a proof checker for this language and a growing database of thousands of proved theorems ; while the Metamath language is not accompanied with an automated theorem prover, it can be regarded as important because the formal language behind it allows development of such a software ; as of March, 2012, there is no " widely " known such software, so it is not a subject of " automated theorem proving " ( it can become such a subject ), but it is a proof assistant.

* The case n = 1 can be illustrated by the claim that there always exist a pair of opposite points on the earth's equator with the same temperature, this can be shown to be true much more easily using the intermediate value theorem.

Since a maximal ideal in A is closed, is a Banach algebra that is a field, and it follows from the Gelfand-Mazur theorem that there is a bijection between the set of all maximal ideals of A and the set Δ ( A ) of all nonzero homomorphisms from A to C. The set Δ ( A ) is called the " structure space " or " character space " of A, and its members " characters.

By a theorem of Gelfand and Naimark, given a B * algebra A there exists a Hilbert space H and an isometric *- homomorphism from A into the algebra B ( H ) of all bounded linear operators on H. Thus every B * algebra is isometrically *- isomorphic to a C *- algebra.

Goldstone's theorem in quantum field theory states that in a system with broken continuous symmetry, there may exist excitations with arbitrarily low energy, called the Goldstone bosons.

* ( Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation ).

The Sylow 2-subgroups are either cyclic, which is easy to handle using the transfer map, or generalized quaternion, which are handled with the Brauer – Suzuki theorem: in particular there are no simple groups of 2-rank 1.

The three classes are groups of GF ( 2 ) type ( classified mainly by Timmesfeld ), groups of " standard type " for some odd prime ( classified by the Gilman – Griess theorem and work by several others ), and groups of uniqueness type, where a result of Aschbacher implies that there are no simple groups.

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.

In recent years there was a high demand for massively distributed databases with high partition tolerance but according to the CAP theorem it is impossible for a distributed system to simultaneously provide consistency, availability and partition tolerance guarantees.

In July of that year, Fermi submitted his doctoral thesis Un teorema di calcolo delle probabilità ed alcune sue applicazioni ( A theorem on probability and some of its applications ) to the Scuola Normale Superiore and received his Laurea from there at the unusually young age of 21.

Furthermore, Bob is only able to perform his measurement once: there is a fundamental property of quantum mechanics, known as the " no cloning theorem ", which makes it impossible for him to make a million copies of the electron he receives, perform a spin measurement on each, and look at the statistical distribution of the results.

Specifically, by the Casorati – Weierstrass theorem, for any transcendental entire function f and any complex w there is a sequence with, is necessarily a polynomial, of degree at least n.

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.

Appel and Haken's approach started by showing that there is a particular set of 1, 936 maps, each of which cannot be part of a smallest-sized counterexample to the four color theorem.

This contradiction means there are no counterexamples at all and that the theorem is therefore true.