supremum norm and L2 nom

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I don't understand why this implies $$ {(\int_{0}^{1}\ { f^{2} dx})^{1/2} \le (\int_{0}^{1}\ { ||f||^{2} dx})^{1/2} =||f||} $$ where || || is a supremum norm. It's from: http://math.uchicago.edu/~may/REU2016/REUPapers/Gaddy.pdf (Theorem 3.6)