In mathematics, an inner product space is a real or complex vector space endowed with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in ⟨ a , b ⟩ {\displaystyle \langle a,b\rangle } . Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or scalar product of Cartesian coordinates. Inner product spaces of infinite dimensions are widely used 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, (denoted | x | {\displaystyle |x|} and | y | {\displaystyle |y|} in the picture); so, every inner product space is a normed vector space. If this normed space is also complete (that is, a Banach space) then the inner product space is a Hilbert space. If an inner product space H is not a Hilbert space, it can be extended by completion to a Hilbert space H ¯ . {\displaystyle {\overline {H}}.} This means that H {\displaystyle H} is a linear subspace of H ¯ , {\displaystyle {\overline {H}},} the inner product of H {\displaystyle H} is the restriction of that of H ¯ , {\displaystyle {\overline {H}},} and H {\displaystyle H} is dense in H ¯ {\displaystyle {\overline {H}}} for the topology defined by the norm.
Definition In this article, F denotes a field that is either the real numbers R , {\displaystyle \mathbb {R} ,} or the complex numbers C . {\displaystyle \mathbb {C} .} A scalar is thus an element of F. A bar over an expression representing a scalar denotes the complex conjugate of this scalar. A zero vector is denoted 0 {\displaystyle \mathbf {0} } for distinguishing it from the scalar 0. An inner product space is a vector space V over the field F together with an inner product, that is, a map
⟨ ⋅ , ⋅ ⟩ : V × V → F {\displaystyle \langle \cdot \operatorname {,} \cdot \rangle :V\times V\to F}
that satisfies the following three properties for all vectors x , y , z ∈ V {\displaystyle x,y,z\in V} and all scalars a , b ∈ F {\displaystyle a,b\in F} .
Conjugate symmetry: ⟨ x , y ⟩ = ⟨ y , x ⟩ ¯ . {\displaystyle \langle x,y\rangle ={\overline {\langle y,x\rangle }}.} As a = a ¯ {\textstyle a={\overline {a}}} if and only if a {\displaystyle a} is real, conjugate symmetry implies that ⟨ x , x ⟩ {\displaystyle \langle x,x\rangle } is always a real number. If F is R {\displaystyle \mathbb {R} } , conjugate symmetry is just symmetry. Linearity in the first argument: ⟨ a x + b y , z ⟩ = a ⟨ x , z ⟩ + b ⟨ y , z ⟩ . {\displaystyle \langle ax+by,z\rangle =a\langle x,z\rangle +b\langle y,z\rangle .}
Positive-definiteness: if x {\displaystyle x} is not zero, then ⟨ x , x ⟩ > 0 {\displaystyle \langle x,x\rangle >0} (conjugate symmetry implies that ⟨ x , x ⟩ {\displaystyle \langle x,x\rangle } is real). If the positive-definiteness condition is replaced by merely requiring that ⟨ x , x ⟩ ≥ 0 {\displaystyle \langle x,x\rangle \geq 0} for all x {\displaystyle x} , then one obtains the definition of positive semi-definite Hermitian form. A positive semi-definite Hermitian form ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is an inner product if and only if for all x {\displaystyle x} , if ⟨ x , x ⟩ = 0 {\displaystyle \langle x,x\rangle =0} then x = 0 {\displaystyle x=\mathbf {0} } .
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