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}).