Theorem: (Royden Real Analysis 4th ed. Section 8.2 Problem 16)

Suppose that E is a measurable set and every function in sight lives in \(L^2(E)\). Further suppose

$$ \lim\int_Ef_n f \to \int_E f^2 = \lim\int_E f_n^2$$

Then \( \{f_n\} \to f\) strongly in \(L^2(E)\).

Proof:

$$ \int_E(f_n-f)^2 = \int_E f_n^2-2\int_E ff_n +\int_Ef^2$$

Which has limit zero by the assumptions of the problem.