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Page "Banach space" ¶ 15
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Theorem and Every
Theorem Every self-adjoint f in A * can be written as f
Scarf ’ s Theorem: Every balanced game has a nonempty core.

Theorem and normed
* Theorem If X is a normed space, then X ′ is a Banach space.
* Theorem Let X be a normed space.

Theorem and space
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.
* 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.

Theorem and is
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: 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: There is a constant c such that
Theorem: K is not a computable function.
He is most famous for proving Fermat's Last Theorem.
: 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.

Theorem and .
The reader will find it helpful to think of the special case when the primes are of degree 1, and even more particularly, to think of the proof of Theorem 10, a special case of this theorem.
Theorem 12.
Since Af are distinct prime polynomials, the polynomials Af are relatively prime ( Theorem 8, Chapter 4 ).
Theorem 13.
Wiles discovered Fermat's Last Theorem on his way home from school when he was 10 years old.
* Lawrence C. Paulson of the University of Cambridge, work on higher-order logic system, co-developer of the Isabelle Theorem Prover
many small primes p, and then reconstructing B < sub > n </ sub > via the Chinese Remainder Theorem.
George Boolos ( 1989 ) built on a formalized version of Berry's paradox to prove Gödel's Incompleteness Theorem in a new and much simpler way.
* Weisstein, Eric W. " Second Fundamental Theorem of Calculus.
: Theorem XXX: " The following classes of partial functions are coextensive, i. e. have the same members: ( a ) the partial recursive functions, ( b ) the computable functions.

Every and reflexive
Every binary relation R on a set S can be extended to a preorder on S by taking the transitive closure and reflexive closure, R < sup >+=</ sup >.
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.
Every finite-dimensional Hausdorff topological vector space is reflexive, because J is bijective by linear algebra, and because there is a unique Hausdorff vector space topology on a finite dimensional vector space.
Every semi-reflexive normed space is reflexive.
Every closed subspace of a reflexive space is reflexive.
Every quotient of a reflexive space is reflexive.
* Every finite-dimensional reflexive algebra is hyper-reflexive.

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