semiinner product
0.0.1 Definition
Let $V$ be a vector space^{} over a field $\mathbb{K}$, where $\mathbb{K}$ is $\mathbb{R}$ or $\u2102$.
A semiinner product on $V$ is a function $\u27e8\cdot ,\cdot \u27e9:V\times V\u27f6\mathbb{K}$ that the following conditions:

1.
$\u27e8{\lambda}_{1}{v}_{1}+{\lambda}_{2}{v}_{2},w\u27e9={\lambda}_{1}\u27e8{v}_{1},w\u27e9+{\lambda}_{2}\u27e8{v}_{2},w\u27e9$ for every ${v}_{1},{v}_{2},w\in V$ and ${\lambda}_{1},{\lambda}_{2}\in \mathbb{K}$.

2.
$\u27e8v,w\u27e9=\overline{\u27e8w,v\u27e9}$ for every $v,w\in V$, where the above means complex conjugation.

3.
$\u27e8v,v\u27e9\ge 0$ ( semi definite).
Hence, a semiinner product on a vector space is just like an inner product^{}, but for which $\u27e8v,v\u27e9$ can be zero ( if $v\ne 0$).
A semiinner product space is just a vector space endowed with a semiinner product.
0.0.2 Topology
0.0.3 CauchySchwarz inequality
The CauchySchwarz inequality is valid for semiinner product spaces:
$$\u27e8v,w\u27e9\le \sqrt{\u27e8v,v\u27e9}\sqrt{\u27e8w,w\u27e9}$$ 
0.0.4 Properties
Let $V$ be a semiinner product space and $W:=\{v\in V:\u27e8v,v\u27e9=0\}$. It is not difficult to see, using the CauchySchwarz inequality, that $W$ is a vector subspace.
The semiinner product in $V$ induces a well defined semiinner product in the quotient (http://planetmath.org/QuotientModule) $V/W$ which is, in fact, an inner product. Thus, the $V/W$ is an inner product space^{}.
Title  semiinner product 
Canonical name  SemiinnerProduct 
Date of creation  20130322 17:47:10 
Last modified on  20130322 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 semidefinite inner product 
Synonym  semi inner product 
Defines  semiinner product space 
Defines  CauchySchwartz inequality for semiinner products 