# inner product space

An inner product space (or pre-Hilbert space) is a vector space (over $\mathbb{R}$ or $\mathbb{C}$) with an inner product ${\langle\cdot,\cdot\rangle}$.

For example, $\mathbb{R}^{n}$ 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 $\|x\|:=\sqrt{{\langle x,\,x\rangle}}$. This norm satisfies the parallelogram law.

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

The Cauchy–Schwarz inequality

 $\displaystyle|{\langle x,\,y\rangle}|\leq\|x\|\cdot\|y\|$ (1)

holds in any inner product space.

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

 $\displaystyle\cos(x,\,y):=\frac{{\langle x,\,y\rangle}}{\|{x}\|\cdot\|{y}\|}.$ (2)

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

 ${\langle x,\,y\rangle}=0.$
 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