# dual basis

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## dual basis

[¦dü·əl ′bā·ses]
(mathematics)
for a finite-dimensional vector space with basis x1, x2, …, xn, the dual basis of the conjugate space is the set of linear functionals f1, f2, …, fn with fi (xi) = 1 and fi (xj) = 0 for i not equal to j.
For a Barach space with basis x 1, x 2, …, the dual basis of the conjugate space is the sequence of continuous linear functionals, f1, f2, …, defined by fi (xi) = 1 and fi (xj) 0 for i not equal to j, provided that the conjugate space is shrinking.
References in periodicals archive ?
FURTHERMORE, if one or more of the nations in the multi-national state had, for a long period of time, lived as geographically separate and autonomous groups, the constitution of the state must, efortiori, be federal, and the constituent states must be organised on the dual basis of language and separate nationality.
Define a morphism [[lambda].sub.M] : Hom([M.sup.*], M) [right arrow] M [cross product] M by [psi] [??] [e.sub.i] [cross product] [psi] ([f.sup.i]),where ([e.sub.i], [f.sup.i]) is a dual basis for M.
Suppose there exists a dual window function that can meet the need of orthogonality, and the dual window constitutes a dual basis, such that
Saudi Arabia has a dual basis strategy towards tackling violence and terrorism on the local and international levels.
In this context, an agreement was signed between the first company under the DTP that is the TNT Business Solutions, the Skills Working Group and the MCCI Business School to develop courses on a dual basis.
"There's a clamor on the DBS side if we can potentially list on a dual basis the FMETF," Dispo said.
So this defines dual Grothendieck polynomials as the dual basis to Grassmannian Grothendieck polynomials.
and [H.sup.a.sub.s] with curl-conforming dual basis functions [[??].sub.j] as
Moreover, the immaculate basis gives rise to a dual basis in the quasi-symmetric function algebra.
The report was released in Bangkok last week, and highlights a decades-long pattern of religious freedom violations, including the destruction of thirteen large Christian crosses, four of them since President Thein Sein's government took power, the construction of fifteen Buddhist pagodas or monasteries by authorities using Chin Christians as forced labour, and more than 40 separate incidents of torture or ill-treatment, targeted at Chin on the dual basis of ethnicity and religion.
Key words and phrases : dual basis, spline-type space, localization.
A dual basis also exists for the draft directive on supplemental pensions (Articles 42 and 94), but the obstacle appears easier to skirt in that case.
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