Let $(V,\lVert\mathord{\cdot}\rVert)$ be a finite-dimensional normed vector space over $k$. A basis $\{e_{i}\}_{i\in\{1,\dots,n\}}$ of $V$ is called orthogonal (with respect to $\lVert\mathord{\cdot}\rVert$) if

$$\forall(c_{1},\dots,c_{r})\in k^{n},\quad\Big\|\sum_{i\in\{1,\dots,n\}}c_{i}e_{i}\Big\|=\max_{i\in\{1,\dots,n\}}\lVert c_{i}e_{i}\rVert$$

Moreover, it is said to be orthonormal if in addition $\lVert e_{i}\rVert=1$ for all $i\in\{1,\dots,n\}$.