semi-inner product

0.0.1 Definition

Let V be a vector spaceMathworldPlanetmath over a field 𝕂, where 𝕂 is or .

A semi-inner product on V is a function ,:V×V𝕂 that the following conditions:

  1. 1.

    λ1v1+λ2v2,w=λ1v1,w+λ2v2,w for every v1,v2,wV and λ1,λ2𝕂.

  2. 2.

    v,w=w,v¯ for every v,wV, where the above means complex conjugation.

  3. 3.

    v,v0 ( semi definite).

Hence, a semi-inner product on a vector space is just like an inner productMathworldPlanetmath, but for which v,v can be zero ( if v0).

A semi-inner product space is just a vector space endowed with a semi-inner product.

0.0.2 Topology

Every semi-inner product space V can be given a topology associated with the semi-inner product. In fact, a semi-norm can be defined in V by


0.0.3 Cauchy-Schwarz inequality

The Cauchy-Schwarz inequality is valid for semi-inner product spaces:


0.0.4 Properties

Let V be a semi-inner product space and W:={vV:v,v=0}. It is not difficult to see, using the Cauchy-Schwarz inequality, that W is a vector subspace.

The semi-inner product in V induces a well defined semi-inner product in the quotient ( V/W which is, in fact, an inner product. Thus, the V/W is an inner product spaceMathworldPlanetmath.

Title semi-inner product
Canonical name SemiinnerProduct
Date of creation 2013-03-22 17:47:10
Last modified on 2013-03-22 17:47:10
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 7
Author asteroid (17536)
Entry type Definition
Classification msc 11E39
Classification msc 15A63
Classification msc 46C50
Synonym positive semi-definite inner product
Synonym semi inner product
Defines semi-inner product space
Defines Cauchy-Schwartz inequality for semi-inner products