Definition An inner product space is a vector space \(V\) over \(K=\RR\) or \(\CC\) equipped with an inner product, \((\cdot,\cdot):V\times V\mapto K\), associating a number \((v,w)\in K\) to every pair of vectors \(v,w\in V\). For \(u,v,w\in V\) and \(a,b\in K\), this inner product must satisfy the linearity condition,
\begin{equation}
(u,av+bw)=a(u,v)+b(u,w),\label{inprod-linear}
\end{equation}
together with one of three possible symmetry properties,
\begin{equation}
(v,w)=(w,v),\label{inprod-symmetric}
\end{equation}
in which case the inner product is called symmetric and the space will be said to have an orthogonal geometry,
\begin{equation}
(v,w)=(w,v)^*,\label{inprod-hermitian}
\end{equation}
in which case the inner product is called Hermitian and the space will be said to have an Hermitian geometry, or
\begin{equation}
(v,w)=-(w,v),\label{inprod-symplectic}
\end{equation}
in which case the inner product is called antisymmetric and the space will be said to have a symplectic geometry.
If the inner product further satisifes the condition,
\begin{equation}
(v,v)\geq 0\text{ and }(v,v)=0 \iff v=0\label{inprod-posdef},
\end{equation}
then it is said to be positive definite, or, if it satisifes the weaker condition,
\begin{equation}
(v,w)=0\:\,\forall v\in V\implies w=0,\label{inprod-non sing}
\end{equation}
then it is said to be non-singular or non-degenerate.

A series of remarks relating to this definition are in order.

Remark When \(K=\CC\) and the inner product is Hermitian, we have,
\begin{equation}
(au+bv,w)=a^*(u,w)+b^*(v,w).
\end{equation}
The inner product is then said to be sequilinear with respect to the first argument. Notice also, that, \((v,v)\in\RR\), \(\forall v\in V\).

Remark In all cases, aside from complex Hermitian, the inner product is bilinear.

Remark When \(K=\RR\), an Hermitian inner product is simply symmetric so when considering Hermitian geometries we only consider vector spaces over \(\CC\).

Remark A negative definite inner product is of course such that \((v,v)\leq0\) and \((v,v)=0\) if and only if \(v=0\).

Remark In a space with symplectic geometry we have \((v,v)=0\), \(\forall v\in V\). In particular, such a space can never be positive (or negative) definite, but may be non-degenerate.

Remark The three symmetry properties described in the definition ensure that \((v,w)=0\) if and only if \((w,v)=0\).

If \(\{e_i\}\) is a basis for \(V\), then we can define a matrix \(\mathbf{G}\) with components \(G_{ij}=(e_i,e_j)\). \(\mathbf{G}\) is called the Gram matrix or matrix of the inner product with respect to the basis \(\{e_i\}\). Symmetric, Hermitian and anti-symmetric inner products then correspond respectively to Gram matrices of that type (recall that \(\mathbf{G}\) is Hermitian if \(\mathbf{G}=\mathbf{G}^\dagger\), where \(\mathbf{G}^\dagger\) is the complex conjugate of the transpose).

If we change basis according to \(e’_i=P_i^je_j\), then \(G’_{ij}=(e’_i,e’_j)=(P_i^ke_k,P_i^le_l)=(P_i^k)^*G_{kl}P_i^l\). So when \(K=\RR\) we have \(\mathbf{G}’=\mathbf{P}^\mathsf{T}\mathbf{G}\mathbf{P}\), and for \(K=\CC\), \(\mathbf{G}’=\mathbf{P}^\dagger\mathbf{G}\mathbf{P}\). In either case the matrices \(\mathbf{G}’\) and \(\mathbf{G}\) are said to be congruent. Note that congruent matrices have the same rank so it makes sense to define the rank of an inner product as the rank of its corresponding Gram matrix in some basis.

Let us consider the property of non-degeneracy in terms of the Gram matrix. Given a basis \(\{e_i\}\) of \(V\), we can define a linear map \(L_\mathbf{G}\) in the usual way such that for any \(v=v^ie_i\in V\), \(L_\mathbf{G}v=\sum_{i,j=1}^nG_{ji}v^ie_j\). Then the rank of \(\mathbf{G}\) is just \(n-\dim\ker L_\mathbf{G}\). A vector \(v\in\ker L_\mathbf{G}\) if and only if \(\sum G_{ji}v^i=0\) for each \(j=1,\dots,n\). But notice that non-degeneracy is equivalent to the statement that \((e_j,v)=0\) for all \(j=1,\dots,n\) implies \(v=0\) that is \(G_{ji}v^j=0\) for each \(j=1,\dots,n\) implies \(v=0\). So we see non-degeneracy is equivalent to \(\ker L_\mathbf{G}\) being trivial, that is, the gram matrix, \(\mathbf{G}\), being of full rank.

A real symmetric inner product space, with a positive definite inner product, is also called a Euclidean vector space.

Example The standard example of a Euclidean vector space is \(\RR^n\) with the inner product of any pair of vectors \(\mathbf{v},\mathbf{w}\in\RR^n\) given by the usual dot or scalar product,
\begin{equation}
(\mathbf{v},\mathbf{w})=\mathbf{v}\cdot\mathbf{w}=\sum_{i=1}^nv^iw^i.
\end{equation}

A real symmetric inner product space with a non-singular inner product is called a pseudo-Euclidean space.

Example The Minkowski space, sometimes denoted, \(\MM^4\), of special relativity is an important example of a pseudo-Euclidean space. It is \(\RR^4\) equipped with the inner product,
\begin{equation}
(v,w)=v^0w^0-\sum_{i=1}^3v^iw^i.
\end{equation}
(Four)-vectors of \(\MM^4\) are conventionally indexed from \(0\), with the 0-component being the ‘time-like’ component and the others being the ‘space-like’ components.

Example \(\CC^n\) has a natural Hermitian geometry when equipped with the inner product defined on any pair of vectors \(v,w\in\CC^n\) by,
\begin{equation}
(v,w)=\sum_{i=1}^n{v^i}^*w^i.
\end{equation}

Example For a simple example of a symplectic geometry on \(K^2\), consider the inner product defined on the standard basis vectors by \((\mathbf{e}_1,\mathbf{e}_2)=1=-(\mathbf{e}_2,\mathbf{e}_1)\) so that for any \(\mathbf{v}\) and \(\mathbf{w}\) in \(K^2\) we have,
\begin{equation}
(\mathbf{v},\mathbf{w})=\det(\mathbf{v},\mathbf{w})=v^1w^2-v^2w^1,
\end{equation}
that is, in the case of \(K=\RR\), the signed area of the parallelogram spanned by \(\mathbf{v}\) and \(\mathbf{w}\).

We’ll see shortly that this symplectic geometry is in fact the only non-degenerate possibility for a 2-dimensional space, up to a notion of equivalence we now define.

Definition An isometry between inner product spaces \(V\) and \(W\) is a linear isomorphism, \(f:V\mapto W\), which preserves the values of the inner products. That is,
\begin{equation}
(u,v)_V=(f(u),f(v))_W,
\end{equation}
for all \(u,v\in V\), where \((\cdot,\cdot)_V\) and \((\cdot,\cdot)_W\) are the inner products on \(V\) and \(W\) respectively. If such an isometry exists, the inner product spaces are said to be isometric.

Remark Clearly, isometric inner product spaces have Gram matrices of the same rank.

Remark Isomorphic spaces equipped with the trivial, zero inner product, are trivially isometric.

Example On \(\RR\), multiplication defines a symmetric inner product, \((x,y)=xy\), which is clearly positive definite. We could also define a negative definite symmetric inner product as, \((x,y)=-xy\). These two inner product spaces cannot be isometric since any automorphism of \(\RR\) is of the form \(f(x)=ax\), \(a\in\RR\), \(a\neq0\), and for \(f\) to be an isometry we’d need, \(x^2=-a^2x^2\), for all \(x\in\RR\). Indeed, this shows us why, on \(\CC\), the inner product spaces with symmetric inner products \((x,y)=xy\) and \((x,y)=-xy\), \(x,y\in\CC\) are isometric – just consider the automorphism \(f(x)=ix\). Staying with \(\CC\), the inner product spaces with Hermitian inner products \((x,y)=x^*y\), the positive definite case, and \((x,y)=-x^*y\), the negative definite case, are not isometric since any automorphism of \(\CC\) is of the form \(f(x)=ax\), \(a\in\CC\), \(a\neq0\), and we’d need, \(\abs{x}^2=-\abs{a}^2\abs{x}^2\), for all \(x\in\CC\).

Finite dimensional vector spaces are classified up to isomorphism in terms of an integer, \(n\), their dimension. In other words two vector spaces are isomorphic if and only if they have the same dimension. Similarly, we would like to classify inner product spaces up to isometry. The dimension will clearly be one of the ‘labels’ of these equivalence classes since in particular isometric spaces are isomorphic. The question is, what other data, related to the inner product, is required to characterise isometric spaces? We’ll see that the key to answering this question lies in expressing a given inner product space as a direct sum of low dimensional ones – the type of inner product determines the structural data needed to characterise the decomposition. We begin, therefore, by considering in detail, low dimensional inner product spaces up to isometry.