inner product
An inner product on a vector space over a field (which must be either the field of real numbers or the field of complex numbers) is a function such that, for all and , the following properties hold:
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1.
(linearity11A small minority of authors impose linearity on the second coordinate instead of the first coordinate.)
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2.
, where denotes complex conjugation (conjugate symmetry)
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3.
, and if and only if (positive definite)
(Note: Rule 2 guarantees that , so the inequality in rule 3 makes sense even when .)
The standard example of an inner product is the dot product on :
Every inner product space is a normed vector space, with the norm being defined by .
Title | inner product |
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Canonical name | InnerProduct |
Date of creation | 2013-03-22 12:13:39 |
Last modified on | 2013-03-22 12:13:39 |
Owner | djao (24) |
Last modified by | djao (24) |
Numerical id | 15 |
Author | djao (24) |
Entry type | Definition |
Classification | msc 11E39 |
Classification | msc 15A63 |
Synonym | Hermitian inner product |
Related topic | InnerProductSpace |
Related topic | HermitianForm |
Related topic | EuclideanVectorSpace |