Definition:Orthonormal Tuple of Elements of Scalar Product Space

Definition

Let $\struct {V, \innerprod \cdot \cdot}$ be a scalar product space.

Let $v_i \in V$ for $i \in \N_{> 0}$.

Let $\tuple {v_1, \ldots, v_k}$ an ordered $k$-tuple.

Then $\tuple {v_1, \ldots v_k}$ is said to be orthonormal if:

$\forall i,j \in \N_{> 0} : i,j \le k : \innerprod {v_i} {v_j} = \pm \delta_{ij}$

where $\delta_{ij}$ is the Kronecker delta.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Pseudo-Riemannian Metrics