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A space X can be embedded into the double dual X ** by

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## Some Related Sentences

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.0.096 seconds.