Everything about Scalar Product totally explained
» For the scalar product or dot product of spatial vectors, see dot product.
In
mathematics, an
inner product space is a
vector space of arbitrary (possibly infinite) dimension with additional
structure, which, among other things, enables generalization of concepts from two or three-dimensional
Euclidean geometry. The additional structure associates to each pair of vectors in the space a number which is called the
inner product (also called a
scalar product) of the vectors. Inner products allow the rigorous introduction of intuitive geometrical notions such as the
angle between vectors or
length of vectors in spaces of all dimensions. It also allows introduction of the concept of
orthogonality between vectors. Inner product spaces generalize
Euclidean spaces (with the
dot product as the inner product) and are studied in
functional analysis.
An inner product space is sometimes also called a
pre-Hilbert space, since its
completion with respect to the
metric,
induced by its inner product, is a
Hilbert space.
Inner product spaces were referred to as
unitary spaces in earlier work, although this terminology is now rarely used.
Definition
In the following article, the
field of
scalars denoted
F is either
the field of
real numbers
R or the field of
complex numbers
C. See below.
Formally, an inner product space is a
vector space V over the field
F together with a
positive-definite sesquilinear form, called an
inner product. For real vector spaces, this is actually a positive-definite
symmetric bilinear form. Thus the inner product is a map
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