i will assume you mean a REAL inner product space (it is not true for a COMPLEX inner product space, there's a different formula).
the norm induced by an inner product <_,_> is given by:
||w|| = √(<w,w>)
if a basis {v_{1},v_{2}} is orthonormal with respect to a given inner product, then:
<v_{1},v_{1}> = <v_{2},v_{2}> = 1
<v_{1},v_{2}> = <v_{2},v_{1}> = 0.
hence, if w = av_{1}+bv_{2}:
||w|| = √(<w,w>) = √(< av_{1}+bv_{2}, av_{1}+bv_{2}>)
= √[<av_{1},av_{1}+bv_{1}> + <bv_{2},av_{1}+bv_{2}>]
= √[<av_{1},av_{1}> + <av_{1},bv_{2}> + <bv_{2},av_{1}> + <bv_{2},bv_{2}>]
= √[a<v_{1},av_{1}> + a<v_{1},bv_{2}> + b<v_{2},av_{1}> + b<v_{2},bv_{2}>]
= √[a^{2}<v_{1},v_{1}> + ab<v_{1},v_{2}> + ab<v_{2},v_{1}> + b^{2}<v_{2},v_{2}>]
= √[a^{2}(1) + ab(0) + ab(0) + b^{2}(1)]
= √(a^{2} + b^{2}).