For a real or complex vector space \({V}\), we can generalize another Cartesian structure, the **inner product** (AKA scalar product, dot product). We define an **inner product space** as including a mapping from vectors to scalars denoted \({\left\langle v,w\right\rangle }\) (also denoted \({(v,w)}\) or \({v\cdot w}\)). The mapping must satisfy:

- \({\left\langle v,w\right\rangle =\left\langle w,v\right\rangle ^{*}}\) where \({^{*}}\) denotes complex conjugation
- \({\left\langle ax+y,bz\right\rangle =a^{*}b\left\langle x,z\right\rangle +b\left\langle y,z\right\rangle }\)
- \({\left\langle v,v\right\rangle >0}\) except if \({v=0}\), in which case it vanishes

The first requirement implies that \({\left\langle v,v\right\rangle }\) is real, and that the inner product is symmetric for real scalars. The second requirement can be phrased as saying that the inner product is **anti-linear** in its first argument and linear in its second, or **sesquilinear**, and the first and second requirements together define a **Hermitian form**. A real inner product is then **bilinear** or **multilinear**, meaning linear in each argument. The third requirement above makes the inner product **positive definite**.

Δ Sometimes the definitions of both inner product and sesquilinear are reversed to make the second argument anti-linear instead of the first. This is sometimes called the “mathematics” convention, while ours would then be the “physics” convention. |

Two vectors are defined to be **orthogonal** if their inner product vanishes. The **orthogonal complement** of a subspace \({W}\) of \({V}\) is the subspace of all vectors orthogonal to every vector in \({W}\), i.e. \({W^{\perp}\equiv\left\{ v\in V\mid\forall w\in W\left\langle v,w\right\rangle =0\right\} }\). A basis of \({W}\) together with a basis for its orthogonal complement \({W^{\perp}}\) forms a basis for all of \({V}\).