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Page "Banach space" ¶ 9
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continuous and dual
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.
The spaceX * of all linear maps into K ( which is called the algebraic dual space to distinguish it from X ′) also induces a weak topology which is finer than that induced by the continuous dual since X ′ ⊆ X *.
There are two types of dual spaces: the algebraic dual space, and the continuous dual space.
When defined for a topological vector space there is a subspace of this dual space, corresponding to continuous linear functionals, which constitutes a continuous dual space.
It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are " enough " continuous linear functionals defined on every normed vector space to make the study of the dual space " interesting.
This theorem establishes an important connection between a Hilbert space and its ( continuous ) dual space: if the underlying field is the real numbers, the two are isometrically isomorphic ; if the field is the complex numbers, the two are isometrically anti-isomorphic.
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
The following theorem, also referred to as the Riesz-Markov theorem, gives a concrete realisation of the dual space of C < sub > 0 </ sub >( X ), the set of continuous functions on X which vanish at infinity.
The term is most commonly used for the initial topology of a topological vector space ( such as a normed vector space ) with respect to its continuous dual.
We may define a possibly different topology on X using the continuous ( or topological ) dual space X < sup >*</ sup >.
The topological dual space consists of all linear functions from X into the base field K which are continuous with respect to the given topology.
If one takes F to be the whole continuous dual space of X, then the weak topology with respect to F coincides with the weak topology defined above.
Specifically, if F is a vector space of linear functionals on X which separates points of X, then the continuous dual of X with respect to the topology σ ( X, F ) is precisely equal to F.
Let X be a normed topological vector space over F, compatible with the absolute value in F. Then in X *, the topological dual space X of continuous F-valued linear functionals on X, all norm-closed balls are compact in the weak -* topology.
* Argentina: 2 years continuous as a permanent resident immediately before the application ( dual citizenship is allowed )
* Belgium: 3 years continuous ( dual citizenship is allowed )< ref >
More generally, a partially defined linear operator A from a topological vector space E into its continuous dual space E < sup >∗</ sup > is said to be symmetric if

continuous and space
** In topology, morphisms between topological spaces are called continuous maps, and an automorphism of a topological space is a homeomorphism of the space to itself, or self-homeomorphism ( see homeomorphism group ).
* 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.
This conservation law mathematically follows from continuous directional symmetry of space ( no direction in space is any different from any other direction ).
In general, a linear mapping on a normed space is continuous if and only if it is bounded on the closed unit ball.
In a Banach space B, the vectors may be notated by kets and the continuous linear functionals by bras.
The prototypical example of a Banach algebra is, the space of ( complex-valued ) continuous functions on a locally compact ( Hausdorff ) space that vanish at infinity.
* The algebra of all bounded continuous real-or complex-valued functions on some locally compact space ( again with pointwise operations and supremum norm ) is a Banach algebra.
* The algebra of all continuous linear operators on a Banach space E ( with functional composition as multiplication and the operator norm as norm ) is a unital Banach algebra.
where is the Gelfand representation of x defined as follows: is the continuous function from Δ ( A ) to C given by The spectrum of in the formula above, is the spectrum as element of the algebra C ( Δ ( A )) of complex continuous functions on the compact space Δ ( A ).
Structures analogous to those found in continuous geometries ( Euclidean plane, real projective space, etc.

continuous and can
In fact, I can only say this seems to me to follow from a wide, continuous, and properly guided exposure to literary art.
What that spirit and attitude were we can best understand if we see more precisely how it contrasts with the communist tradition with the longest continuous history, the one which reached Christianity by the way of Stoicism through the Church Fathers of Late Antiquity.
Also available is a slitter which `` peels '' the inside of a folded block of foam and can be used to slit continuous sheets up to 300 yd. in length, down to 1/16 in. thick.
Watch people flock to amusement houses, cocktail lounges, and night clubs that advertise continuous entertainment, which means an endless flow of noise and frivolity by paid entertainers who are supposed to perform in those incredible ways which are designed to give men a few hours of dubious relaxation -- watch them and you can tell that many of them are running away from something.
They take special care not to destroy the termite mound or consume the entire colony, which ensures that the termites can rebuild and provide a continuous supply of food.
It differs from a digital signal, in which a continuous quantity is represented by a discrete function which can only take on one of a finite number of values.
By using the palette to mask the top area of the display and taking care about when it changes mode it can shift the continuous graphics at the bottom of the display down in two pixel increments because the internal display counter is not incremented on blank scanlines during non-continuous graphics modes.
The Knudsen number can be used to guide the choice between statistical mechanics and the continuous formulation of aerodynamics.
Beam bridges are horizontal beams supported at each end by substructure units and can be either simply supported when the beams only connect across a single span, or continuous when the beams are connected across two or more spans.
Although the show typically broadcasts daily updates in the evening ( which are sometimes criticized for their heavy editing ), viewers can also watch a continuous, 24-hour feed from multiple cameras on the web.
A function from the set of real numbers to the real numbers can be represented by a graph in the Cartesian plane ; the function is continuous if, roughly speaking, the graph is a single unbroken curve with no " holes " or " jumps ".
These definitions are equivalent to one another, so the most convenient definition can be used to determine whether a given function is continuous or not.
For a time-dependent process of heating of the calorimetric material, defined by a continuous joint progression of and, starting at time and ending at time, there can be calculated an accumulated quantity of heat delivered,.
This implies that the partial derivatives of u and v exist ( although they need not be continuous ) and we can approximate small variations of f linearly.
The process can culminate in corona discharges, streamers, leaders, or in a spark or continuous electric arc that completely bridges the gap.
Chimps may also hunt as a coordinated team, so that they can corner their prey even in a continuous canopy.
Although digital representations are discrete, the information represented can be either discrete, such as numbers and letters or continuous, such as sounds, images, and other measurements.
Both batch and continuous distillations can be improved by making use of a fractionating column on top of the distillation flask.
Sketch drawings use similar techniques, although with pencils and drawing sticks continuous variations in tone can be achieved.
* It can also provide uniformly spaced samples of the continuous DTFT of a finite length sequence.
From a classical perspective, the electromagnetic field can be regarded as a smooth, continuous field, propagated in a wavelike manner ; whereas from the perspective of quantum field theory, the field is seen as quantized, being composed of individual particles.
In " The Meaning of Relativity ," Einstein wrote, " One can give good reasons why reality cannot at all be represented by a continuous field.

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