If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

If Af is the null space of Af, then Theorem 12 says that Af.

Theorem: If K < sub > 1 </ sub > and K < sub > 2 </ sub > are the complexity functions relative to description languages L < sub > 1 </ sub > and L < sub > 2 </ sub >, then there is a constant c – which depends only on the languages L < sub > 1 </ sub > and L < sub > 2 </ sub > chosen – such that

Theorem 1: If a property is positive, then it is consistent, i. e., possibly exemplified.

Theorem 2: If something is God-like, then the property of being God-like is an essence of that thing.

If on the other hand Theorem 2 holds and φ is valid in all structures, then ¬ φ is not satisfiable in any structure and therefore refutable ; then ¬¬ φ is provable and then so is φ, thus Theorem 1 holds.

If we also include the German speaking Vienna, during the Weimar years Mathematician Kurt Gödel published his groundbreaking Incompleteness Theorem.

If n is not a prime power, then every Sylow subgroup is proper, and, by Sylow's Third Theorem, we know that the number of Sylow p-subgroups of a group of order n is equal to 1 modulo p and divides n. Since 1 is the only such number, the Sylow p-subgroup is unique, and therefore it is normal.

Theorem: If R is a filtered ring whose associated graded ring gr ( R ) is a domain, then R itself is a domain.

If we consider measuring distance based on Pythagoras ' Theorem then it is clear that we shall be adding quantities measured in different units, and so this leads to meaningless results.

* ( Kronecker's Theorem ) If p is an irreducible monic integer polynomial with, then either p ( z )=

Theorem ( Gregoire de Saint-Vincent 1647 ) If bc

If now diameter AF is drawn bisecting DC so that DF and CF are sides c of an inscribed decagon, Ptolemy's Theorem can again be applied – this time to cyclic quadrilateral ADFC with diameter d as one of its diagonals:

:: If, then by Fermat's Little Theorem each of the numbers is congruent to one modulo.

: Theorem: If S is any set then S cannot contain elements of all cardinalities.

* Theorem Let X be a normed space.

: Theorem on projections: Let the function f: X → B be such that a ~ b → f ( a )

:: Theorem ( Lefschetz theorem on ( 1, 1 )- classes ) Any element of H < sup > 2 </ sup >( X, Z ) ∩ H < sup > 1, 1 </ sup >( X ) is the cohomology class of a divisor on X.

The Relative Hurewicz Theorem states that if each of X, A are connected and the pair ( X, A ) is ( n − 1 )- connected then H < sub > k </ sub >( X, A ) = 0 for k < n and H < sub > n </ sub >( X, A ) is obtained from π < sub > n </ sub >( X, A ) by factoring out the action of π < sub > 1 </ sub >( A ).

The Triadic Hurewicz Theorem states that if X, A, B, and C = A ∩ B are connected, the pairs ( A, C ), ( B, C ) are respectively ( p − 1 )-, ( q − 1 )- connected, and the triad ( X ; A, B ) is p + q − 2 connected, then H < sub > k </ sub >( X ; A, B ) = 0 for k < p + q − 2 and H < sub > p + q − 1 </ sub >( X ; A ) is obtained from π < sub > p + q − 1 </ sub >( X ; A, B ) by factoring out the action of π < sub > 1 </ sub >( A ∩ B ) and the generalised Whitehead products.

Theorem B is sharp in the sense that if H < sup > 1 </ sup >( X, F ) = 0 for all coherent sheaves F on a complex manifold X ( resp.

* Although Sard's Theorem does not hold in general, every continuous map f: X → R < sup > n </ sup > from a Hilbert manifold can be arbitrary closely approximated by a smooth map g: X → R < sup > n </ sup > which has no critical points

In the notation of the proof of Theorem 12, let us take a look at the special case in which the minimal polynomial for T is a product of first-degree polynomials, i.e., the case in which each Af is of the form Af.

By Theorem 10, D is a diagonalizable operator which we shall call the diagonalizable part of T.

Theorem: There is a constant c such that

Theorem: K is not a computable function.

He is most famous for proving Fermat's Last Theorem.

* Theorem Every reflexive normed space is a Banach space.

: Turing's thesis: " Turing's thesis that every function which would naturally be regarded as computable is computable under his definition, i. e. by one of his machines, is equivalent to Church's thesis by Theorem XXX.

One particularly important physical result concerning conservation laws is Noether's Theorem, which states that there is a one-to-one correspondence between conservation laws and differentiable symmetries of physical systems.

: Theorem ( A. Korselt 1899 ): A positive composite integer is a Carmichael number if and only if is square-free, and for all prime divisors of, it is true that ( where means that divides ).

Even though the text is otherwise inferior to the 1621 edition, Fermat's annotations — including the " Last Theorem "— were printed in this version.

( This is the Fundamental Theorem of Equivalence Relations, mentioned above );

Image: Thales ' Theorem Simple. svg | Thales ' theorem: if AC is a diameter, then the angle at B is a right angle.

The identity of is unique by Theorem 1. 4 below.

* James ` s Theorem For a Banach space the following two properties are equivalent:

Theorem: Let V be a topological vector space

Urysohn's Theorem can be restated as: A topological space is separable and metrizable if and only if it is regular, Hausdorff and second-countable.

* A space elevator is also constructed in the course of Clarke's final novel ( co-written with Frederik Pohl ), The Last Theorem.

Liouville's Theorem shows that, for conserved classical systems, the local density of microstates following a particle path through phase space is constant as viewed by an observer moving with the ensemble ( i. e., the total or convective time derivative is zero ).

# The First Fundamental Theorem of Asset Pricing: A discrete market on a discrete probability space ( Ω,, ) is arbitrage-free if and only if there exists at least one risk neutral probability measure that is equivalent to the original probability measure, P.

Theorem ( Fuglede ) Let T and N be bounded operators on a complex Hilbert space with N being normal.

Theorem ( Calvin Richard Putnam ) Let T, M, N be linear operators on a complex Hilbert space, and suppose that M and N are normal and MT

* Green's Theorem, one of several theorems that connect an integral in n-dimensional space with one in ( n − 1 )- dimensional space

Theorem: Let V be a finite-dimensional vector space over a field F, and A a square matrix over F. Then V ( viewed as an F-module with the action of x given by A and extending by linearity ) satisfies the F-module isomorphism

* T. R. Johansen, The Bochner-Minlos Theorem for nuclear spaces and an abstract white noise space, 2003.