inner product space

An inner product space (or pre-Hilbert space) is a vector space (over $ℝ$ or $ℂ$) with an inner product $⟨\cdot ,\cdot ⟩$.

For example, ${ℝ}^n$ with the familiar dot product forms an inner product space.

Every inner product space is also a normed vector space, with the norm defined by $\parallel x\parallel :=\sqrt{⟨x,x⟩}$. This norm satisfies the parallelogram law.

If the metric $\parallel x-y\parallel$ induced by the norm is complete (http://planetmath.org/Complete), then the inner product space is called a Hilbert space.

The Cauchy–Schwarz inequality

$|⟨x,y⟩|\le \parallel x\parallel \cdot \parallel y\parallel$ (1)

holds in any inner product space.

According to (1), one can define the angle between two non-zero vectors $x$ and $y$:

$\cos (x,y):={\displaystyle \frac{⟨x,y⟩}{\parallel x\parallel \cdot \parallel y\parallel }}.$ (2)

This provides that the scalars are the real numbers. In any case, the perpendiculatity of the vectors may be defined with the condition

$$⟨x,y⟩=0.$$

Title	inner product space
Canonical name	InnerProductSpace
Date of creation	2013-03-22 12:14:05
Last modified on	2013-03-22 12:14:05
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	23
Author	CWoo (3771)
Entry type	Definition
Classification	msc 46C99
Synonym	pre-Hilbert space
Related topic	InnerProduct
Related topic	OrthonormalBasis
Related topic	HilbertSpace
Related topic	EuclideanVectorSpace2
Related topic	AngleBetweenTwoLines
Related topic	FluxOfVectorField
Related topic	CauchySchwarzInequality
Defines	angle between two vectors
Defines	perpendicularity