The context: We are looking at orthogonality and general Fourier series. Given $\lambda$ an eigenvalue, and $w$ the corresponding eigenfunction, we are studying the eigenvalue problem: $w'' = -\lambda w$, subject to some Boundary Conditions.
My question is why can we assume in that: $||w||^2 = 1$ $\iff$ $\int_{a}^{b} (w(x))^2 dx = 1 $? This was given in a proof, assumed without loss of generality
The equation you are looking at is linear; this means in particular that a multiple of a solution is again a solution. Depending on what you are doing having a norm-one solution could be useful (say you had computations where $\|w\|$ appears, having all those norms replaced by $1$ certainly simplify things).
Of course you cannot make that assumption completely for free: if for example you have a non-zero initial condition, multiplying a solution by a number does not give you another solution. But if your initial condition is zero, or if you are not considering initial conditions, you can always replace $w$ with $w/\|w\|$.