Orthonormal family

Definition

Let $V$ be an inner product space over $\R$ or $\C$, and let $I$ be a set. A family $(e_i)_{i \in I}$ of vectors of $V$ is orthonormal when, for all $i, j \in I$, $$\langle e_i, e_j \rangle = \delta_{ij},$$ where $\delta_{ij} = 1$ if $i = j$ and $\delta_{ij} = 0$ if $i \ne j$.

Remarks

So each $e_i$ has norm $1$, and $e_i$ and $e_j$ are orthogonal for $i \ne j$. The standard basis of $\R^n$ is orthonormal for the dot product. The functions $t \mapsto e^{2\pi i k t}$, $k \in \Z$, are orthonormal for $\langle f, g \rangle = \int_0^1 f \bar g\, dt$.

Explanations

L1 human

An orthonormal family is like a set of coordinate axes: unit vectors at right angles. Computations with such vectors are simple, because all inner products are $0$ or $1$. They are the best bases to work with in an inner product space.