Vector weighting functions

Orthogonal but not Orthonormal Basis Sets

Suppose we have two vectors from an orthonormal system, $\overrightarrow{\mathbf{u}}$ and $\overrightarrow{\mathbf{v}}$. Taking the inner product of these vectors, we get

$\overrightarrow{\mathbf{u}}\bullet \overrightarrow{\mathbf{v}}=\sum _{k=1}^3{u}_k{\overrightarrow{\mathbf{a}}}_k\bullet \sum _{m=1}^3{v}_m{\overrightarrow{\mathbf{a}}}_m=\sum _{k=1}^3{u}_k\sum _{m=1}^3{v}_m{\overrightarrow{\mathbf{a}}}_k\bullet {\overrightarrow{\mathbf{a}}}_m=\sum _{k=1}^3{u}_k\sum _{m=1}^3{v}_m{\delta }_{k,m}=\sum _{k=1}^3{v}_k{u}_k$

What if they aren't from a normalized system, so that