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'Reflexive non-degenerate sesquilinear'
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| Title of object: |
Reflexive non-degenerate sesquilinear |
| Canonical Name: |
ReflexiveNonDegenerateSesquilinear |
| Type: |
Definition |
| Created on: |
2006-04-14 19:28:37 |
| Modified on: |
2006-06-19 13:05:13 |
| Classification: |
msc:15A63 |
| Keywords: |
Reflexive |
| Defines: |
Reflexive non-degenerate sesquilinear, Reflexive non-degenerate bilinear, Reflexive |
| Synonyms: |
Reflexive non-degenerate sesquilinear=reflexive non-degenerate bilinear
Reflexive non-degenerate sesquilinear=reflexive sesquilinear
Reflexive non-degenerate sesquilinear=reflexive bilinear |
Revision comment (for changes between this and next version):
| Changes for correction #9531 ('capitalization'). |
Preamble:
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Content:
A non-degenerate sesquilinear form $b:V\times V\rightarrow k$ is \emph{reflexive} if for all $v,w\in V$, if $b(v,w)=0$ then $b(w,v)=0$. This means
\[v\perp w\textnormal{ if and only if } w\perp v.\]
It is rare to define perpendicularity for sesquilinear/bilinear maps which are not reflexive because it would require a version of left and right perpendicular. Thus a reflexive sesquilinear/bilinear map is usually synonymous with the existence of perpendicularity. |
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