Orthonormal with respect to an inner product

Question

What does it mean for a basis to be orthonormal with respect to an inner product?

Answer 1

if you have vectors (or functions) $u,v,w$ you must have

$u.u=1$, $v.v=1$, $w.w=1$

$u.v=0$, $v.w=0$, $w.u=0$

Remember that the order of the dot product is unimportant

Answer 2

$\{v_1,v_2,...v_n\}$ is said to be Orthonormal basis if

• $<v_i,v_j> = 0$ for $i\neq j$
• $<v_i,v_j>$ = $1$ for $i = j$.