Let be a vector space over a field , where is or .
A semi-inner product on is a function that the following conditions:
for every and .
for every , where the above means complex conjugation.
( semi definite).
Hence, a semi-inner product on a vector space is just like an inner product, but for which can be zero ( if ).
A semi-inner product space is just a vector space endowed with a semi-inner product.
0.0.3 Cauchy-Schwarz inequality
The Cauchy-Schwarz inequality is valid for semi-inner product spaces:
Let be a semi-inner product space and . It is not difficult to see, using the Cauchy-Schwarz inequality, that is a vector subspace.
|Date of creation||2013-03-22 17:47:10|
|Last modified on||2013-03-22 17:47:10|
|Last modified by||asteroid (17536)|
|Synonym||positive semi-definite inner product|
|Synonym||semi inner product|
|Defines||semi-inner product space|
|Defines||Cauchy-Schwartz inequality for semi-inner products|