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Page "Tychonoff space" ¶ 34
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space and X
* Given any Banach space X, the continuous linear operators A: XX 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.
Thus, the vector space B ( X, Y ) can be given the operator norm
With respect to this norm B ( X, Y ) is a Banach space.
This is also true under the less restrictive condition that X be a normed space.
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 is
Experience is not seen, as it is in classical rationalism, as presenting us initially with clear and distinct objects simply located in space and registering their character, movements, and changes on the tabula rasa of an uninvolved intellect.
It is notable that at this time he was writing with admiration of Cimabue's and Poussin's way of filling space.
Next I refer to our program in space exploration, which is often mistakenly supposed to be an integral part of defense research and development.
Third, the United States is pressing forward in the development of large rocket engines to place vehicles of many tons into space for exploration purposes.
Like ours, the economy of the space merchants must constantly expand in order to survive, and, like ours, it is based on the principle of `` ever increasing everybody's work and profits in the circle of consumption ''.
And this, of course, is exactly what Madison Avenue has been accused of doing albeit in a primitive way, with its `` hidden persuaders '' and what the space merchants accomplish with much greater sophistication and precision.
Thus the copywriter in the world of the space merchants is the person who in earlier ages might have been a lyric poet, the person `` capable of putting together words that stir and move and sing ''.
But there is hope, for Conservation Commissioner Bontempo has tagged the sanctuary as the kind of place the state hopes to include in its program to double its park space.
Four hundred million dollars of the increase is for the expanded space program, a responsibility similarly neglected by Mr. Eisenhower.
I would like to see you devote some space in an early issue to the news blackout concerning President Kennedy's activities, so far as Southern California is concerned.
Advice is given also on problems of plant location and plant space.
The outside 4-inch space is filled by mortaring blocks on edge.
The space between them is filled with pit-run gravel or earth.
In general, such apartments afford more protection than smaller buildings because their walls are thick and there is more space.
The problem for the city apartment dweller is primarily to plan the use of existing space.
The music is always allowed the living space needed to attain its full sonority ; ;
But if space and money are no problem and small children are not on hand every day, it is certainly more restful to have your pool and entertainment area removed from the immediate environs of the house.
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.
With detectors sensitive to three mass intervals and based on a few counts, the second and third Russian space probes indicate that the flux of the smallest particles detected is less than that of larger ones.
Of course, if there is a dust blanket around the Earth, the fluxes in interplanetary space should be less than the figures given here.
If ( remember this is an assumption ) the minimal polynomial for T decomposes Af where Af are distinct elements of F, then we shall show that the space V is the direct sum of the null spaces of Af.
Conversely, suppose that **ya is in the null space of Af.

space and completely
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.
Modeling an object as a continuum assumes that the substance of the object completely fills the space it occupies.
In phase space, equivalent commutators of function star-products are called Moyal brackets, and are completely isomorphic to the Hilbert-space commutator structures mentioned.
With a traditional optical telescope, the space between stars and galaxies ( the background ) is completely dark.
Raw CMBR data coming down from the space vehicle ( i. e., WMAP ) contain foreground effects that completely obscure the fine-scale structure of the Cosmic Microwave background.
As a result of their rearing, Kal-El grows up to think of himself as Clark Kent, completely unaware of his alien heritage until he was well into adulthood ( the fact that the Kents found him sometime just after the Cold War meant that the Kents assumed the rocket they found him in was from an Earth-based space program-possibly that of Communist Russia-and that Clark had superhuman abilities because he was either a metahuman or some kind of genetic experiment ).
** Solid-core-Can consist of low-density particle board or foam used to completely fill the space within the door.
** Solid-core-Can consist of low-density particle board or foam used to completely fill the space within the door.
Hilbert spaces can be completely classified: there is a unique Hilbert space up to isomorphism for every cardinality of the orthonormal basis.
Eventually both galaxies will have merged completely, streams of gas and dust will be flying through the space near the newly formed giant elliptical galaxy.
, all GUT models which aim to be completely realistic are quite complicated, even compared to the Standard Model, because they need to introduce additional fields and interactions, or even additional dimensions of space.
Under meager point-set hypotheses, namely sobriety, this is completely accurate — it is possible to recover a sober space from its associated site.
Microsoft had a double gain from its release: first, it made it impossible for consumers to run Windows 95 on a cheaper, non-Microsoft DOS ; secondly, although traces of DOS were never completely removed from the system and MS DOS 7 would be loaded briefly as a part of the booting process, Windows 95 applications ran solely in 386 enhanced mode, with a flat 32-bit address space and virtual memory.
To every Lie group, we can associate a Lie algebra, whose underlying vector space is the tangent space of G at the identity element, which completely captures the local structure of the group.
The second Mars Habitat Unit flies to Mars with the crew, its interior given over completely to living and storage space.
Two liquids of different densities completely fill a narrow space between two sheets of plexiglass.
* Plenum ( physics ), a space completely filled with matter
* Reduced row echelon form, a certain reduced row echelon form of a matrix which completely and uniquely determines its row space
This can be reduced somewhat by using a binary space partitioning tree to reduce the amount of time spent determining which patches are completely hidden from others in complex scenes ; but even so, the time spent to determine the form factor still typically scales as n log n. New methods include adaptive integration
* Equivalence principle — Einstein was able to show that the effects of gravity were completely equivalent to a rocket's acceleration in any small region of space
X is a completely regular space if given any closed set F and any point x that does not belong to F, then there is a continuous function f from X to the real line R such that f ( x ) is and, for every y in F, f ( y ) is.
X is a Tychonoff space, or T < sub > 3½ </ sub > space, or T < sub > π </ sub > space, or completely T < sub > 3 </ sub > space if it is both completely regular and Hausdorff.

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