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Difference between revisions of "Co-basis"

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A [[Basis|basis]] of a dual or [[Adjoint space|adjoint space]].
 
A [[Basis|basis]] of a dual or [[Adjoint space|adjoint space]].
  
 
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A co-basis is also called a dual basis. Often the terminology  "dual basis"  or  "co-basis"  of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022680/c0226801.png" /> is used to denote a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022680/c0226802.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022680/c0226803.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022680/c0226804.png" /> (the Kronecker delta), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022680/c0226805.png" /> is a given basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022680/c0226806.png" />. The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022680/c0226807.png" /> is referred to as a pair of dual bases.
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A co-basis is also called a dual basis. Often the terminology  "dual basis"  or  "co-basis"  of $E^*$ is used to denote a basis $(e_i^*)$ of $E^*$ such that $e_i^*(e_j)=\delta_{ij}$ (the Kronecker delta), where $(e_i)$ is a given basis of $E$. The pair $(e_i),(e_i^*)$ is referred to as a pair of dual bases.

Revision as of 18:30, 21 October 2014

A basis of a dual or adjoint space.

Comments

A co-basis is also called a dual basis. Often the terminology "dual basis" or "co-basis" of $E^*$ is used to denote a basis $(e_i^*)$ of $E^*$ such that $e_i^*(e_j)=\delta_{ij}$ (the Kronecker delta), where $(e_i)$ is a given basis of $E$. The pair $(e_i),(e_i^*)$ is referred to as a pair of dual bases.

How to Cite This Entry:
Co-basis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Co-basis&oldid=33980