Testing The Real Inner Product Axioms

Three Conditions to Check

An operation may resemble a familiar dot product without satisfying the rules that make an inner product useful. To test one, separate the requirements: symmetry compares the two orders of its inputs, linearity controls how it responds to sums and scalar multiples, and positive definiteness checks its value when the two inputs are equal. These are independent checks: passing one or two does not guarantee that the operation is an inner product.

Let \(V\) be a vector space over \(\mathbb{R}\). A proposed real-valued operation is a function \(B:V\times V\to\mathbb{R}\). Linearity in the first slot means that, for all \(\mathbf{u},\mathbf{v},\mathbf{w}\in V\) and \(c\in\mathbb{R}\), it satisfies both additivity and homogeneity:

For a real inner product, symmetry and linearity in the first slot also give linearity in the second slot. This is useful when checking a formula: it is enough to verify linearity in one slot, provided symmetry has been checked too.

Proposition. If \(B:V\times V\to\mathbb{R}\) is symmetric and linear in its first slot, then it is linear in its second slot.

Proof. Let \(\mathbf{u},\mathbf{v},\mathbf{w}\in V\) and \(c\in\mathbb{R}\). For additivity in the second slot, symmetry, first-slot additivity, and symmetry again give \(B(\mathbf{u},\mathbf{v}+\mathbf{w})=B(\mathbf{v}+\mathbf{w},\mathbf{u})=B(\mathbf{v},\mathbf{u})+B(\mathbf{w},\mathbf{u})=B(\mathbf{u},\mathbf{v})+B(\mathbf{u},\mathbf{w})\). For homogeneity, symmetry and first-slot homogeneity give \(B(\mathbf{u},c\mathbf{v})=B(c\mathbf{v},\mathbf{u})=cB(\mathbf{v},\mathbf{u})=cB(\mathbf{u},\mathbf{v})\). These are the two required properties. \(\square\)

A Matrix Test in \(\mathbb{R}^n\)

On \(\mathbb{R}^n\), write vectors as columns. A matrix \(\mathbf{A}\in\mathbb{R}^{n\times n}\) defines an operation \(B(\mathbf{u},\mathbf{v})=\mathbf{u}^{\top}\mathbf{A}\mathbf{v}\). The matrix test below turns the symmetry and linearity checks into an algebraic condition. Positive definiteness still requires checking the associated quadratic expression for every nonzero vector.

Proof. Suppose first that \(B\) is an inner product. For the standard coordinate vectors \(\mathbf{e}_i,\mathbf{e}_j\), we have \(B(\mathbf{e}_i,\mathbf{e}_j)=a_{ij}\) and \(B(\mathbf{e}_j,\mathbf{e}_i)=a_{ji}\). Symmetry gives \(a_{ij}=a_{ji}\) for every \(i,j\), so \(\mathbf{A}^{\top}=\mathbf{A}\). Positive definiteness applied to any nonzero \(\mathbf{x}\) gives \(\mathbf{x}^{\top}\mathbf{A}\mathbf{x}=B(\mathbf{x},\mathbf{x})>0\).

Conversely, suppose \(\mathbf{A}^{\top}=\mathbf{A}\) and \(\mathbf{x}^{\top}\mathbf{A}\mathbf{x}>0\) whenever \(\mathbf{x}\ne\mathbf{0}\). Matrix multiplication is linear, so for any \(\mathbf{u},\mathbf{v},\mathbf{w}\in\mathbb{R}^n\) and \(c\in\mathbb{R}\), \(B(\mathbf{u}+\mathbf{v},\mathbf{w})=B(\mathbf{u},\mathbf{w})+B(\mathbf{v},\mathbf{w})\) and \(B(c\mathbf{u},\mathbf{v})=cB(\mathbf{u},\mathbf{v})\). Also, \(B(\mathbf{u},\mathbf{v})=\mathbf{u}^{\top}\mathbf{A}\mathbf{v}=(\mathbf{u}^{\top}\mathbf{A}\mathbf{v})^{\top}=\mathbf{v}^{\top}\mathbf{A}^{\top}\mathbf{u}=\mathbf{v}^{\top}\mathbf{A}\mathbf{u}=B(\mathbf{v},\mathbf{u})\). Finally, the assumed quadratic inequality gives positive definiteness for every nonzero vector, and \(B(\mathbf{0},\mathbf{0})=0\). Thus all the inner product conditions hold. \(\square\)

Worked Checks in \(\mathbb{R}^2\)

A Reliable Testing Order

For a formula on \(\mathbb{R}^n\), testing the conditions in a fixed order makes the reasoning efficient. A single failure is enough to conclude that the operation is not an inner product; there is no need to establish that it passes every other test. Conversely, to establish that it is an inner product, all three conditions must be verified for arbitrary vectors, not just for a sample pair.

A common mistake is to treat a positive diagonal expression as sufficient. For example, \(C(\mathbf{u},\mathbf{v})=(u_1v_1+u_2v_2)^2\) is symmetric and has \(C(\mathbf{x},\mathbf{x})>0\) for every nonzero \(\mathbf{x}\). The tempting but incorrect conclusion is that \(C\) is an inner product. The correction is to test linearity too: \(C(2\mathbf{e}_1,\mathbf{e}_1)=4\), while \(2C(\mathbf{e}_1,\mathbf{e}_1)=2\). Since these are unequal, homogeneity fails.

For matrix-defined operations, another tempting shortcut is to inspect only the diagonal entries of \(\mathbf{A}\). The matrix test requires symmetry of the whole matrix and positivity of \(\mathbf{x}^{\top}\mathbf{A}\mathbf{x}\) for every nonzero \(\mathbf{x}\); checking a few entries or vectors does not establish those requirements. The examples above separate the issues: one operation fails positivity despite symmetry, another fails linearity despite symmetry and positivity on the diagonal, and another has positive diagonal values but fails symmetry.

Why the Axioms Matter

The inner product conditions make the operation consistent across all vectors and scalars, rather than merely producing plausible values for a few inputs. Symmetry rules out dependence on the order of the pair. Linearity ensures that combinations of vectors behave predictably under the operation. Positive definiteness prevents a nonzero vector from having a zero or negative self-pairing. When these conditions hold, the operation is a real inner product; when one fails, identifying the failed condition gives a precise reason the proposed formula does not qualify.