Inner Product Space
An Inner Product Space is a vector space where … Inner Product/Inner Dot Product ...
- AKA: Pre-Hilbert Space, Dot Product Space.
- Context:
- Euclidean Space ...
- See: Cosine Similarity Distance, Dot Product, Canonical Dot Product.
References
2011
- (Wikipedia, 2011) ⇒ http://en.wikipedia.org/wiki/Inner_product
- In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors. Inner products allow the rigorous introduction of intuitive geometrical notions such as the length of a vector or the angle between two vectors. They also provide the means of defining orthogonality between vectors (zero inner product). Inner product spaces generalize Euclidean spaces (in which the inner product is the dot product, also known as the scalar product) to vector spaces of any (possibly infinite) dimension, and are studied in functional analysis.
An inner product naturally induces an associated norm, thus an inner product space is also a normed vector space. A complete space with an inner product is called a Hilbert space. An incomplete space with an inner product is called a pre-Hilbert space, since its completion with respect to the norm, induced by the inner product, becomes a Hilbert space. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces.
- In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors. Inner products allow the rigorous introduction of intuitive geometrical notions such as the length of a vector or the angle between two vectors. They also provide the means of defining orthogonality between vectors (zero inner product). Inner product spaces generalize Euclidean spaces (in which the inner product is the dot product, also known as the scalar product) to vector spaces of any (possibly infinite) dimension, and are studied in functional analysis.
2009
- Chi Woo. (2009). http://planetmath.org/encyclopedia/InnerProductSpace.html
- QUOTE: An inner product space (or pre-Hilbert space) is a vector space (over [math]\displaystyle{ \mathbb{R} }[/math] or [math]\displaystyle{ \mathbb{C} }[/math]) with an inner product [math]\displaystyle{ {\langle ∙,∙\rangle} }[/math]. For example, [math]\displaystyle{ \mathbb{R}^n }[/math] 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 [math]\displaystyle{ \Vert x \Vert := \sqrt{ {\langle x,\,x\rangle}} }[/math]. This norm satisfies the parallelogram law. If the metric [math]\displaystyle{ \Vert{x-y}\Vert }[/math] induced by the norm is complete, then the inner product space is called a Hilbert space. The Cauchy-Schwarz inequality [math]\displaystyle{ \displaystyle \vert{\langle x,\,y\rangle}\vert \le \Vert x\Vert \cdot\Vert y\Vert }[/math] holds in any inner product space.