In mathematics, an inner product space or a Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors, often denoted using angle brackets (as in
⟨
a
,
b
⟩
{\displaystyle \langle a,b\rangle }
). 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. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898.An inner product naturally induces an associated norm, (
|
x
|
{\displaystyle |x|}
and
|
y
|
{\displaystyle |y|}
are the norms of
x
{\displaystyle x}
and
y
{\displaystyle y}
in the picture), which canonically makes every inner product space into a normed vector space. If this normed space is also a Banach space then the inner product space is called a Hilbert space. If an inner product space
(
H
,
⟨
⋅
,
⋅
⟩
)
{\displaystyle (H,\langle \,\cdot \,,\,\cdot \,\rangle )}
is not a Hilbert space then it can be "extended" to a Hilbert space
(
H
¯
,
⟨
⋅
,
⋅
⟩
H
¯
)
,
{\displaystyle \left({\overline {H}},\langle \,\cdot \,,\,\cdot \,\rangle _{\overline {H}}\right),}
called a completion. Explicitly, this means that
H
{\displaystyle H}
is linearly and isometrically embedded onto a dense vector subspace of
H
¯
{\displaystyle {\overline {H}}}
and that the inner product
⟨
⋅
,
⋅
⟩
H
¯
{\displaystyle \langle \,\cdot \,,\,\cdot \,\rangle _{\overline {H}}}
on
H
¯
{\displaystyle {\overline {H}}}
is the unique continuous extension of the original inner product