The continuous dual space can be used to define a new topology on X: the weak topology.

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 *.

* Corollary If X is a Banach space, then X is reflexive if and only if X ′ is reflexive, which is the case if and only if its unit ball is compact in the weak topology.

Equipped with the topology of pointwise convergence on A ( i. e., the topology induced by the weak -* topology of A < sup >∗</ sup >), the character space, Δ ( A ), is a Hausdorff compact space.

* On the other hand, the closed unit ball of the dual of a normed space is compact for the weak -* topology.

The analytic techniques used for studying Lie groups must be replaced by techniques from algebraic geometry, where the relatively weak Zariski topology causes many technical complications.

In mathematics, weak topology is an alternative term for initial topology.

) with respect to the weak topology.

) with respect to the weak topology.

The weak topology on X is the initial topology

In order to distinguish the weak topology from the original topology on X, the original topology is often called the strong topology.

A subbase for the weak topology is the collection of sets of the form φ < sup >- 1 </ sup >( U ) where φ ∈ X < sup >*</ sup > and U is an open subset of the base field K. In other words, a subset of X is open in the weak topology if and only if it can be written as a union of ( possibly infinitely many ) sets, each of which is an intersection of finitely many sets of the form φ < sup >- 1 </ sup >( U ).

More generally, if F is a subset of the algebraic dual space, then the initial topology of X with respect to F, denoted by σ ( X, F ), is the weak topology with respect to F.

The UN army is too weak, too demoralized for the task.

It is no harder to raise big, healthy, blooming plants than weak, sickly little things ; ;

If, however, the child is weak, overweight, or afraid, more help will be needed.

Analysis in roleplaying is usually done for the purpose of understanding strong and weak points of an individual or as a process to eliminate weak parts and strengthen good parts.

weak hands ask, What is going on??

Pimen is an old man, weak in body -- his voice rarely rises to a full forte -- but firm and clear of mind.

The fear of war can make us either too weak to stand and too willing to compromise, or too reckless and too nervous to negotiate for peace as long as there is any chance to negotiate.

" At its worst, Scheler says, " love for the small, the poor, the weak, and the oppressed is really disguised hatred, repressed envy, an impulse to detract, etc., directed against the opposite phenomena: wealth, strength, power, largesse.

Some species in the Plethodontidae have a weak zone at the base of the tail and if the salamander is in danger or if the tail is grabbed by a predator, it breaks off, a process known as autotomy.

The gases are contained in a test-tube ( A ) standing over a large quantity of weak alkali ( B ), and the current is conveyed in wires insulated by U-shaped glass tubes ( CC ) passing through the liquid and round the mouth of the test-tube.

Because of weak bonding between the layers, gray arsenic is brittle and has a relatively low Mohs hardness of 3. 5.

The side-chain can make an amino acid a weak acid or a weak base, and a hydrophile if the side-chain is polar or a hydrophobe if it is nonpolar.

It receives its name from its soil ( weathered from the weak underlying limestone ), which is black in colour, almost destitute of sand and loam, and rich in limestone and marl formations, especially adapted to the production of cotton ; hence the region is also called the Cotton Belt.

Though the resolution is historically significant, Hideaki Uemura, professor at Keisen University in Tokyo and a specialist in indigenous peoples ' rights, commented that the motion is " weak in the sense of recognizing historical facts " as the Ainu were " forced " to become Japanese in the first place.

Tramadol is structurally closer to venlafaxine than to codeine and delivers analgesia by not only delivering " opiate-like " effects ( through mild agonism of the mu receptor ) but also by acting as a weak but fast-acting serotonin releasing agent and norepinephrine reuptake inhibitor.

Flupirtine is a centrally acting K + channel opener with weak NMDA antagonist properties.

Harmine and harmaline are selective and reversible inhibitors of monoamine oxidase A ( MAO-A ), while tetrahydroharmine is a weak serotonin reuptake inhibitor ( SRI ).

However, many have argued that the usefulness of these laws is vastly inadequate in terms of controlling government actions, largely because of institutional and systemic obstacles like a weak judiciary, poorly trained judges and lawyers, and corruption.

By 454, the Delian League could be fairly characterized as an Athenian Empire ; at the start of the Peloponnesian War, only Chios and Lesbos were left to contribute ships, and these states were by now far too weak to secede without support.

ABC News characterized public consensus on Clinton as, " You can't trust him, he's got weak morals and ethics and he's done a heck of a good job.

* Woven bone, which is characterized by haphazard organization of collagen fibers and is mechanically weak

A Grand Unified Theory, ( GUT ), is a model in particle physics in which at high energy, the three gauge interactions of the Standard Model which define the electromagnetic, weak, and strong interactions, are merged into one single interaction characterized by one larger gauge symmetry and thus one unified coupling constant.

The governments of Papua New Guinea are characterized by weak political parties and highly unstable parliamentary coalitions.

Solomon Islands governments are characterized by weak political parties and highly unstable parliamentary coalitions.

They defined homology and cohomology as functors equipped with natural transformations subject to certain axioms ( e. g., a weak equivalence of spaces passes to an isomorphism of homology groups ), verified that all existing ( co ) homology theories satisfied these axioms, and then proved that such an axiomatization uniquely characterized the theory.

In statistical sense, a time series r < sub > t </ sub > is characterized as having weak white noise if

During the ' 60s and ' 70s, the Pistons were characterized by very strong individuals and weak teams.

Rhythm in music is characterized by a repeating sequence of stressed and unstressed beats ( often called " strong " and " weak ") and divided into bars organized by time signature and tempo indications.

The BIOS are characterized as stealthy super-soldiers, and have the lowest signature, making them hard to detect, to counter their comparatively weak hull.

Hsp70 by itself is characterized by a very weak ATPase activity, such that spontaneous hydrolysis will not occur for many minutes.

Overall, during his time in office Beackon was characterized as a weak councillor who had trouble following the council agenda.

The Fy-x allelle is characterized by a weak anti-Fy-b reaction and appears to be the result of two separate transitions: Cytosine265Threonine ( Arginine89Cysteine ) and Guanine298Adenosine ( Alanine100Threonine ).

The attraction is created by an electronic transition into an excited electronic state, and is best characterized as a weak electron resonance.

Trichorrhexis nodosa is a defect in the hair shaft characterized by thickening or weak points ( nodes ) that cause the hair to break off easily.

In the weak field approximation where interaction between fields is not taken into account, NGT is characterized by a symmetric rank-2 tensor field ( gravity ), an antisymmetric tensor field, and a constant characterizing the mass of the antisymmetric tensor field.

Considered by many writers too weak and good-natured to cope with the problems which confronted him, Agis was characterized by a sincerity of purpose and a blend of youthful modesty with royal dignity, which render him perhaps the most attractive figure in the whole of Spartan history.

Account for inactivity of Seleucus IV Philopator in Syria, who was characterized as weak and pacific, might be consisted in part by war exhaustion of Syria in this period

The plateau areas above the Cane Creek Gorge are characterized by poor soil and weak resource potential, both exacerbated by the area's limited accessibility ( by the 1920s, no major railroads and one crude highway passed between Pikeville and Spencer ).

He characterized the treaty as " humanitarian in purpose, but impracticable in operation ", and believed the league would be weak.

In the years following the death of Sierra Leone ’ s first prime minister Sir Milton Margai in 1964, politics in the country was increasingly characterized by corruption, mismanagement, and electoral violence that led to a weak civil society, the collapse of the education system, and, by 1991, an entire generation of dissatisfied youth, members of which were attracted to the rebellious message of the Revolutionary United Front ( RUF ).

His philosophy can be characterized as postmodern with his emphasis on " pensiero debole " ( weak thought ).