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A space X can be embedded into the double dual X ** by
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space and X
* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.
* Given any topological space X, the continuous real-or complex-valued functions on X form a real or complex unitary associative algebra ; here the functions are added and multiplied pointwise.
Is X a Banach space, the space B ( X ) = B ( X, X ) forms a unital Banach algebra ; the multiplication operation is given by the composition of linear maps.
If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.
space and can
From the brightness of the F component of the solar corona and the brightness of the zodiacal light, an estimate of the particle sizes, concentrations, and spatial distribution can be derived for regions of space near the ecliptic plane.
But first, we must define two terms so that their meaning will be clearly understood: form -- any unique sequence of alphabetic characters that can appear in a language preceded and followed by a space ; ;
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.
Could your future, your immediate future, be made to include taking us upstairs, giving us a bit of space in which our friend can work, and making available to him your finger tips ''??
About that same time John Crosby's TV series on the popular arts proved again that giving jazz ample breathing space is one of the most sensible things a producer can do.
While Renaissance artists sought nature to find their style, the Mannerists looked first for a style and found a manner. In Mannerist paintings, compositions can have no focal point, space can be ambiguous, figures can be characterized by an athletic bending and twisting with distortions, exaggerations, an elastic elongation of the limbs, bizarre posturing on one hand, graceful posturing on the other hand, and a rendering of the heads as uniformly small and oval.
Precise measurements of the relative positions of the inner planets can be made by radar and by telemetry from space probes.
Then we can express the variables as the sum of the ( time averaged ) mean field () that varies in space and a small fluctuating field () that varies in space and time.
Alpha decay can provide a safe power source for radioisotope thermoelectric generators used for space probes and artificial heart pacemakers.
The violation of one's personal space or touching in a way the victim deemed inappropriate can be simple assault.
This system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates ( x, y, z ).
Animals can respond to extreme heat, for example, through natural heat acclimation or by burrowing into the ground to find a cooler space.
By sweeping this surface through R < sup > 3 </ sup > as a function of the ion sequence input data, such as via ion-ordering, a volume is generated onto which positions the 2D detector positions can be computed and placed three-dimensional space.
Watterson opposed the structure publishers imposed on Sunday newspaper cartoons: the standard cartoon starts with a large, wide rectangle featuring the cartoon's logo or a throwaway panel tangential to the main area so that newspapers pressed for space can remove the top third of the cartoon if they wish ; the rest of the strip is presented in a series of rectangles of different widths.
space and be
Next I refer to our program in space exploration, which is often mistakenly supposed to be an integral part of defense research and development.
The Industrial Division published, in 1960, a new, attractive industrial brochure, `` Rhode Island -- Right For Industry '', and prepared copy for a new edition of the Directory Of Manufacturers ( to be printed shortly ), and for a new space catalogue.
Areas should be large enough to include the attractions, have ample space for the use of facilities needed, and have room around the edges to protect the values of the area from encroachment by private developments.
The sweep of space, the delicate counterbalance of the white masses, the over-all completeness and unity, the originality and imagination, all entitle it to be called an authentic masterpiece.
Under this assumption the space group must be Af and the following are the positions of the atoms in the unit cell.
Since there is a continual loss of micrometeoritic material in space because of the radiation effects, there must be a continual replenishment: otherwise, micrometeorites would have disappeared from interplanetary space.
Of course, if there is a dust blanket around the Earth, the fluxes in interplanetary space should be less than the figures given here.
Second, even if the characteristic polynomial factors completely over F into a product of polynomials of degree 1, there may not be enough characteristic vectors for T to span the space V.
Let V be a finite-dimensional vector space over an algebraically closed field F, e.g., the field of complex numbers.
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.
Let N be a positive integer and let V be the space of all N times continuously differentiable functions F on the real line which satisfy the differential equation Af where Af are some fixed constants.
In their very first collages, Braque and Picasso draw or paint over and on the affixed paper or cloth, so that certain of the principal features of their subjects as depicted seem to thrust out into real, bas-relief space -- or to be about to do so -- while the rest of the subject remains imbedded in, or flat upon, the surface.
If, on the other hand, they opted for representation, it had to be representation per se -- representation as image pure and simple, without connotations ( at least, without more than schematic ones ) of the three-dimensional space in which the objects represented originally existed.