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If a point in the Hilbert cube is specified by a sequence with, then a homeomorphism to the infinite dimensional unit cube is given by.
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If and point
If he had been `` liquidated '' in some way, he would have become a martyr, a rallying point for people who shared his ideas.
If we examine the three types of change from the point of view of their internal structure we find an additional profound difference between the third and the first two, one that accounts for the notable difference between the responses they evoke.
If he were to go with White, he would be out there two days, not just listening in the dark at some point between here and Papa-san, but moving ever deeper into enemy land -- behind Papa-san -- itself.
If their schedules were to synchronize, there was no point in wasting time.
If design head has a deep cavity, clay lid will be quite thick at this point ; ;
If the argument is accepted as essentially sound up to this point, it remains for us to consider whether the patient's difficulties in orienting himself spatially and in locating objects in space with the sense of touch can be explained by his defective visual condition.
If this attitude is seriously questioned in the Soviet Union, it does not necessarily follow that the majority of the society in which I live is too aware of the necessity for clarity on this ethical as well as aesthetic point of view.
If the Southerners were sufficiently aroused, they could very well cut the Kennedy legislative program to ribbons from their vantage point of committee chairmanships, leaving Sam Rayburn leading a truncated, unworkable party.
If it was, then it must have been God's intention to translate him at a certain point from time to eternity.
If one also removes the second postulate (" a line can be extended indefinitely ") then elliptic geometry arises, where there is no parallel through a point outside a line, and in which the interior angles of a triangle add up to more than 180 degrees.
If the object point be infinitely distant, all rays received by the first member of the system are parallel, and their intersections, after traversing the system, vary according to their perpendicular height of incidence, i. e. their distance from the axis.
If the object point O is infinitely distant, u1 and u2 are to be replaced by h1 and h2, the perpendicular heights of incidence ; the sine condition then becomes sin u ' 1 / h1 = sin u ' 2 / h2.
If, in an unsharp image, a patch of light corresponds to an object point, the center of gravity of the patch may be regarded as the image point, this being the point where the plane receiving the image, e. g., a focusing screen, intersects the ray passing through the middle of the stop.
If a system consists of several particles, the total angular momentum about a point can be obtained by adding ( or integrating ) all the angular momenta of the constituent particles.
If the net force on some body is directed always toward some fixed point, the center, then there is no torque on the body with respect to the center, and so the angular momentum of the body about the center is constant.
If the score reaches 20-all, then the game continues until one side gains a two point lead ( such as 24 – 22 ), up to a maximum of 30 points ( 30 – 29 is a winning score ).
If the filter shows amplitude ripple within the passband, the x dB point refers to the point where the gain is x dB below the nominal passband gain rather than x dB below the maximum gain.
If the player and dealer have the same point total, this is called a " push " and the player typically doesn't win or lose money on that hand.
If the heat of vaporization and the vapor pressure of a liquid at a certain temperature is known, the normal boiling point can be calculated by using the Clausius-Clapeyron equation thus:
If the large-scale Universe appears isotropic as viewed from Earth, the cosmological principle can be derived from the simpler Copernican principle, which states that there is no preferred ( or special ) observer or vantage point.
If a player is one point away from winning a match, that player's opponent will always want to double as early as possible in order to catch up.
If the holes are closed, the gas will only mix with ambient air at the point of combustion, that is, only after it has exited the tube at the top.
If and Hilbert
If the norm of a Banach space satisfies this identity, the associated inner product which makes it into a Hilbert space is given by the polarization identity.
If a system is composed of two subsystems described in V and W respectively, then the Hilbert space of the entire system is the tensor product of the two spaces.
If the earlier completion procedure is applied to a normed vector space, the result is a Banach space containing the original space as a dense subspace, and if it is applied to an inner product space, the result is a Hilbert space containing the original space as a dense subspace.
If the underlying manifold is allowed to be infinite dimensional ( for example, a Hilbert manifold ), then one arrives at the notion of an infinite-dimensional Lie group.
If ρ acts on a finite dimensional Hilbert space and has eigenvalues
Let H be a Hilbert space, and let H * denote its dual space, consisting of all continuous linear functionals from H into the field R or C. If x is an element of H, then the function φ < sub > x </ sub >, defined by
In 1900, David Hilbert posed an influential question about transcendental numbers, Hilbert's seventh problem: If a is an algebraic number, that is not zero or one, and b is an irrational algebraic number, is a < sup > b </ sup > necessarily transcendental?
If R is a ring, let R denote the ring of polynomials in the indeterminate X over R. Hilbert proved that if R is " not too large ", in the sense that if R is Noetherian, the same must be true for R. Formally,
* If R is a Noetherian ring, then R is Noetherian by the Hilbert basis theorem.
If the kernel used is a Gaussian radial basis function, the corresponding feature space is a Hilbert space of infinite dimensions.
Hilbert himself declared: " If I were to awaken after having slept for a thousand years, my first question would be: has the Riemann hypothesis been proven?
If a normal operator on a finite-dimensional real or complex Hilbert space ( inner product space ) stabilizes a subspace, then it also stabilizes its orthogonal complement.
If X is a Hilbert space and T is a normal operator, then a remarkable result known as the spectral theorem gives an analogue of the diagonalisation theorem for normal finite-dimensional operators ( Hermitian matrices, for example ).
If A has a multiplicative identity 1, then it is immediate that the equivalence class ξ in the GNS Hilbert space H containing 1 is a cyclic vector for the above representation.
If X is the toric variety corresponding to the normal fan of P, then P defines an ample line bundle on X, and the Ehrhart polynomial of P coincides with the Hilbert polynomial of this line bundle.
If one wants to consider antipodal points as identified, one passes to projective space ( see also projective Hilbert space, for this idea as applied in quantum mechanics ).
If, then the distinguished vector in the Hilbert space is thought of as the vacuum state defined by.
If is a bounded linear operator on a Hilbert space, then this notion coincides with the condition that
If is a quadratic irrational, then the j-invariant is an algebraic integer of degree, the class number of and the minimal ( monic integral ) polynomial it satisfies is called the Hilbert class polynomial.
If one thinks of operators on a Hilbert space as " generalized complex numbers ", then the adjoint of an operator plays the role of the complex conjugate of a complex number.
Tentative Proof: If the underlying Hilbert space is finite-dimensional, the spectral theorem says that N is of the form
If it does not converge, is no longer well-defined, but a modified definition where one integrates over arbitrarily large, relatively compact domains, still yields the Einstein equation as the Euler – Lagrange equation of the Einstein – Hilbert action.
If A < sub > 1 </ sub >, ..., A < sub > n </ sub > are n operators each localized in a bounded region and U ( a ) represents the unitary operator actively translating the Hilbert space by the vector a, then if we pick some subset of the n operators to translate,
If U is an isometric map defined on a closed subset H < sub > 1 </ sub > of a Hilbert space H then we can define an extension W of U to all of H by the condition that W be zero on the orthogonal complement of H < sub > 1 </ sub >.
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