semi-inner product

0.0.1 Definition

Let $V$ be a vector space over a field $𝕂$, where $𝕂$ is $ℝ$ or $ℂ$.

A semi-inner product on $V$ is a function $⟨\cdot ,\cdot ⟩:V\times V⟶𝕂$ that the following conditions:

1. $⟨{\lambda }_1{v}_1+{\lambda }_2{v}_2,w⟩={\lambda }_1⟨v_1,w⟩+{\lambda }_2⟨v_2,w⟩$ for every $v_1,v_2,w\in V$ and ${\lambda }_1,{\lambda }_2\in 𝕂$.

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

3. $⟨v,v⟩\ge 0$ ( semi definite).

Hence, a semi-inner product on a vector space is just like an inner product, but for which $⟨v,v⟩$ can be zero ( if $v\ne 0$).

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 $\parallel \cdot \parallel$ can be defined in $V$ by

$$\parallel v\parallel :=\sqrt{⟨v,v⟩}$$

0.0.3 Cauchy-Schwarz inequality

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

$$|⟨v,w⟩|\le \sqrt{⟨v,v⟩}\sqrt{⟨w,w⟩}$$

0.0.4 Properties

Let $V$ be a semi-inner product space and $W:=\{v\in V:⟨v,v⟩=0\}$. 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 (http://planetmath.org/QuotientModule) $V/W$ which is, in fact, an inner product. Thus, the $V/W$ is an inner product space.

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