Strong convergence in $L^2(E)$ if this limit and integral exchange

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.

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categories / real analysis theorem
tags / lp spaces, strong convergence, convergence, weak convergence, bounded linear functionals, functional analysis, real analysis

Strong convergence in \(L^2(E)\) if this limit and integral exchange