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**NonCommAlg** this is a special case of Holder's inequality. however, it can be easily proved as an inequality on its own: clearly for any real number $\displaystyle \lambda$ we have: $\displaystyle \int_0^1(\lambda - |f(x)|)^2 \ dx \geq 0,$ which gives us:

$\displaystyle p(\lambda)=\lambda^2 - \left(2\int_0^1 |f(x)| \ dx \right) \lambda + \int_0^1 |f(x)|^2 \ dx \geq 0, \ \ \forall \lambda \in \mathbb{R}.$ thus the discriminant of the quadratic function $\displaystyle p(\lambda)$ must be $\displaystyle \leq 0,$ that is: $\displaystyle \left(\int_0^1 |f(x)| \ dx \right)^2 \leq \int_0^1 |f(x)|^2 \ dx. \ \ \Box$