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Theorem and Let
* Theorem Let X be a normed space.
: Theorem on projections: Let the function f: X → B be such that a ~ b → f ( a )
Theorem: Let R be a Dedekind domain with fraction field K. Let L be a finite degree field extension of K and denote by S the integral closure of R in L. Then S is itself a Dedekind domain.
Theorem: Let R be a Dedekind domain.
Theorem: Let A ∈ C < sup > n × n </ sup > be a complex-valued matrix and ρ ( A ) its spectral radius ; then
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
Theorem: Let T be a bounded linear operator from to and at the same time from to.
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
Theorem of Oka: Let M be a complex manifold,

Theorem and V
1990b, Review of V. A. Uspensky, Gödel's Incompleteness Theorem, Journal of Symbolic Logic 55: 889-891.
* The old axiom V. 2 is now Theorem 32.
Theorem V. 4. 5, p. 156
An example is the following result ( Fröhlich and Taylor, Chapter V, Theorem 1. 25 ).

Theorem and be
: 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.
This result is now known as Church's Theorem or the Church – Turing Theorem ( not to be confused with the Church – Turing thesis ).
Fine, Do Correlations need to be explained ?, in Philosophical Consequences of Quantum Theory: Reflections on Bell's Theorem, edited by Cushing & McMullin ( University of Notre Dame Press, 1986 ).
In a ring all of whose ideals are principal ( a principal ideal domain or PID ), this ideal will be identical with the set of multiples of some ring element d ; then this d is a greatest common divisor of a and b. But the ideal ( a, b ) can be useful even when there is no greatest common divisor of a and b. ( Indeed, Ernst Kummer used this ideal as a replacement for a gcd in his treatment of Fermat's Last Theorem, although he envisioned it as the set of multiples of some hypothetical, or ideal, ring element d, whence the ring-theoretic term.
It was then simplified in 1947, when Leon Henkin observed in his Ph. D. thesis that the hard part of the proof can be presented as the Model Existence Theorem ( published in 1949 ).
The Model Existence Theorem and its proof can somehow be formalized in the framework of PA.
By the Chinese Remainder Theorem, this ring factors into the direct product of rings of integers mod p. Now each of these factors is a field, so it is clear that the only idempotents will be 0 and 1.
In the General Possibility Theorem, Kenneth Arrow argues that if a legislative consensus can be reached through a simple majority, then minimum conditions must be satisfied, and these conditions must provide a superior ranking to any subset of alternative votes ( Arrow 1963 ).
~ p ∨ p. Since p → p is true ( this is Theorem 2. 08, which is proved separately ), then ~ p ∨ p must be true.
#* Note: This fact provides a proof of the infinitude of primes distinct from Euclid's Theorem: if there were finitely many primes, with p being the largest, we reach an immediate contradiction since all primes dividing 2 < sup > p </ sup > − 1 must be larger than p .</ li >
This theorem was established by John von Neumann, who is quoted as saying " As far as I can see, there could be no theory of games … without that theorem … I thought there was nothing worth publishing until the Minimax Theorem was proved ".
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.
Using Rogers ' characterization of acceptable programming systems, Rice's Theorem may essentially be generalized from Turing machines to most computer programming languages: there exists no automatic method that decides with generality non-trivial questions on the black-box behavior of computer programs.
On 13 August, 2012, this project was officially announced to be The Zero Theorem, set to start shooting in Bucharest on October 22, produced by Dean Zanuck ( son to the late Richard D. Zanuck who was to originally produce in 2009 ), worldwide sales handled by Voltage Pictures, Toronto and starring Academy Award winner Christoph Waltz in the lead, replacing Billy Bob Thornton who had been attached to the project in 2009.
On the other hand, the scholars invoking Gödel's Theorem appear, at least in some cases, to be referring not to the underlying rules, but to the understandability of the behavior of all physical systems, as when Hawking mentions arranging blocks into rectangles, turning the computation of prime numbers into a physical question.
Where the angle is a right angle, also known as the Hypotenuse-Leg ( HL ) postulate or the Right-angle-Hypotenuse-Side ( RHS ) condition, the third side can be calculated using the Pythagoras ' Theorem thus allowing the SSS postulate to be applied.
The max-flow min-cut theorem is a special case of the duality theorem for linear programs and can be used to derive Menger's theorem and the König-Egerváry Theorem.
Theorem: The angle may be trisected if and only if is reducible over the field extension Q.

Theorem and topological
* Dylan G. L. Allegretti, Simplicial Sets and van Kampen's Theorem ( Discusses generalized versions of van Kampen's theorem applied to topological spaces and simplicial sets ).
Allegretti, Simplicial Sets and van Kampen's Theorem ( Discusses generalized versions of van Kampen's theorem applied to topological spaces and simplicial sets ).
The 2-dimensional Gauss – Bonnet Theorem arises as the special case where the analytical index is defined in terms of Betti numbers and the topological index is defined in terms of the Gauss – Bonnet integrand.

Theorem and vector
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.
Recall that Hamiltonian vector fields are divergencefree in even Poisson geometry because of Liouville's Theorem.
For such objects, the integral may be taken over the entire surface () by taking the absolute value of the integrand ( so that the " top " and " bottom " of the object do not subtract away, as would be required by the Divergence Theorem applied to the constant vector field ) and dividing by two:
Theorem 1 demonstrates that where an orbiting body is subject only to a centripetal force, it follows that a radius vector, drawn from the body to the attracting center, sweeps out equal areas in equal times ( no matter how the centripetal force varies with distance ).
Theorem 4 shows that with a centripetal force inversely proportional to the square of the radius vector, the time of revolution of a body in an elliptical orbit with a given major axis is the same as it would be for the body in a circular orbit with the same diameter as that major axis.

Theorem and space
If Af is the null space of Af, then Theorem 12 says that Af.
* Theorem If X is a normed space, then X ′ is a Banach space.
* Theorem Every reflexive normed space is a Banach space.
* James ` s Theorem For a Banach space the following two properties are equivalent:
* 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.
* Green's Theorem, one of several theorems that connect an integral in n-dimensional space with one in ( n − 1 )- dimensional space
* T. R. Johansen, The Bochner-Minlos Theorem for nuclear spaces and an abstract white noise space, 2003.

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